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\begin{paragr}\label{paragr:defoograph}
  An \emph{$\oo$-graph} $C$ consists of a sequence $(C_k)_{k \in \mathbb{N}}$ of sets together with maps
  \[ \begin{tikzcd}
    C_{k-1} &\ar[l,"s",shift left] \ar[l,"t"',shift right] C_{k}
  \end{tikzcd}
  \]
  for every $k > 0$, subject to the \emph{globular identities}:
  \begin{equation*}
  \left\{
  \begin{aligned}
    s \circ s &= s \circ t, \\
    t \circ t &= t \circ s.
  \end{aligned}
  \right.
  \end{equation*}
  Elements of $C_k$ are called \emph{$k$-cells} or \emph{cells of dimension $k$}.
  For $x$ a $k$-cell with $k>0$, $s(x)$ is the \emph{source} of $x$ and $t(x)$ is the \emph{target} of $x$.
  
  More generally for all $k<n \in \mathbb{N}$, we define maps $s_k,t_k : C_n \to C_k$ as
  \[
  s_k = \underbrace{s\circ \dots \circ s}_{n-k \text{ times}}
  \]
  and
    \[
  t_k = \underbrace{t\circ \dots \circ t}_{n-k \text{ times}}.
  \]
  For an $n$-cell $x$, $s_k(x)$ is the \emph{$k$-source} of $x$ and $t_k(x)$ the \emph{$k$-target} of $x$.


  Two $k$-cells $x$ and $y$ are \emph{parallel} if $k=0$ or $k>0$ and
  \[
  s(x)=s(y) \text{ and } t(x)=t(y).
  \]
   For all $k<n \in \mathbb{N}$, we define the set $C_n\underset{C_k}{\times}C_n$ as the following fibred product
    \[
    \begin{tikzcd}
      C_n\underset{C_k}{\times}C_n \ar[r] \ar[dr,phantom,"\lrcorner", very near start] \ar[d] &C_n \ar[d,"t_k"]\\
      C_n \ar[r,"s_k"] & C_k.
      \end{tikzcd}
    \]
    That is, elements of $C_n\underset{C_k}{\times}C_n$ are pairs $(x,y)$ of $n$-cells such that $s_k(x)=t_k(y)$. We say that two $n$-cells $x$ and $y$ are \emph{$k$-composable} if the pair $(x,y)$ belongs to $C_n\times_{C_k}C_n$. 
\end{paragr}

\begin{paragr}\label{paragr:defmoroograph}
    Let $C$ and $C'$ be two $\oo$-graphs. A \emph{morphism of $\oo$-graphs} $f : C \to C'$ from $C$ to $C'$ is a sequence $(f_k : C_k \to D_k)_{k \in \mathbb{N}}$ of maps such that for every $k>0$, the squares
    \[
    \begin{tikzcd}
    C_k \ar[d,"s"] \ar[r,"f_k"]&C'_k \ar[d,"s"] \\
    C_{k-1} \ar[r,"f_{k-1}"] & C'_{k-1}
    \end{tikzcd}
\quad
        \begin{tikzcd}
    C_k \ar[d,"t"] \ar[r,"f_k"]&C'_k \ar[d,"t"] \\
    C_{k-1} \ar[r,"f_{k-1}"] & C'_{k-1}
    \end{tikzcd}
    \]
    are commutative.

    For a $k$-cell $x$, we will often write $f(x)$ instead of $f_k(x)$.
    We denote by $\ooGrph$ the category of $\oo$-graphs and morphisms of $\oo$-graphs.
\end{paragr}
\begin{paragr}
  For $n \in \mathbb{N}$, the notion of \emph{$n$-graph} is defined similarly, only this time there is only a finite sequence $(C_k)_{0 \leq k \leq n}$ of cells.
  For example, a $0$-graph is just a set and a $1$-graph is an ordinary graph
  \[
  \begin{tikzcd}
    C_0 & \ar[l,shift right,"t"'] \ar[l,shift left,"s"] C_1.
    \end{tikzcd}
  \]
  The definition of morphism of $n$-graphs is the same as for $\omega$-graphs, only this time there is only a finite sequence $(f_k : C_k \to C'_k)_{0 \leq k \leq n}$ of maps. We denote by $\nGrph$ the category of $n$-graphs and morphisms of $n$-graphs.  
\end{paragr}
\begin{paragr}
  Let $n \in \nbar$. An \emph{$n$-magma} consists of:
  \begin{itemize}
  \item[-] an $n$-graph $C$,
  \item[-] maps
    \[
    \begin{aligned}
    (\shortminus)\underset{k}{\ast}(\shortminus)  : C_l\underset{C_k}{\times}C_l &\to C_l \\
      (x,y) &\mapsto x\underset{k}{\ast}y
      \end{aligned}
    \]
    for all $l,k \in \mathbb{N}$ with $k < l \leq n$,\footnote{Note that if $n=\omega$, then $l<n$ because we supposed that $l \in \mathbb{N}$.}
  \item[-] maps
    \[
    \begin{aligned}
    1_{(\shortminus)} : C_k &\to C_{k+1}\\
      x &\mapsto 1_x
      \end{aligned}
    \]
    for every $k \in \mathbb{N}$ with $k\leq n$,
  \end{itemize}
  subject to the following axioms:
  \begin{itemize}
  \item[-] for all $k,l \in \mathbb{N}$ with $k<l\leq n$ and every $k$-composable $l$-cells $x$ and $y$,
    \[
    s(x\underset{k}{\ast} y) =
    \begin{cases}
      s(y) &\text{ when }k=l-1,\\
      s(x)\underset{k}{\ast} s(y) &\text{ otherwise,}
      \end{cases}
    \]
    and
        \[
    t(x\underset{k}{\ast} y) =
    \begin{cases}
      t(x) & \text{ when }k=l-1,\\
      t(x)\underset{k}{\ast} t(y) &\text{ otherwise.}
      \end{cases}
    \]
  \item[-]for every $k \in \mathbb{N}$ with $k\leq n$ and every $k$-cell,
    \[
    s(1_x)=t(1_x)=x.
    \]
  \end{itemize}
  We will use the same letter to denote an $n$-magma and its underlying $n$-graph.
  
   For two $k$-composable $l$-cells $x$ and $y$, we refer to $x\ast_ky$ as the \emph{$k$-composition} of $x$ and $y$.
   For a $k$-cell $x$, we refer to $1_{x}$ as the \emph{unit on $x$}.
   
      \remtt{Je n'aime pas trop les notations et définitions des unités itérées qui suivent.}
    More generally, for any $l \in \mathbb{N}$ with $k < l\leq n$, we define $\1^{l}_{(\shortminus)} : C_k \to C_l$ as
    \[
    \1^{l}_{(\shortminus)} := \underbrace{1_{(\shortminus)} \circ \dots \circ 1_{(\shortminus)}}_{l-k \text{ times }} : C_k \to C_l.
    \]
    Let $x$ be a $k$-cell, $\1^l_x$ is the \emph{$l$-dimensional unit on $x$}, and for consistency, we also set
    \[
    \1^{k}_x := x.
    \]
    A cell is \emph{degenerate} if it is a unit on a strictly lower dimensional cell. 
\end{paragr}
\begin{paragr}
  Let $n \in \nbar$ and let $C$ and $C'$ be $n$-magmas. A \emph{morphism of $n$-magmas} $f : C \to C'$ is a morphism of $n$-graphs that is compatible with compositions and units. This means:
  \begin{itemize}
  \item[-]for all $k,l \in \mathbb{N}$ with $k<l\leq n$ and every $k$-composable $l$-cells $x$ and $y$,
    \[
    f(x\underset{k}{\ast}y)=f(x)\underset{k}{\ast}f(y),
    \]
  \item[-]for every $k \in \mathbb{N}$ with $k\leq n$ and every $k$-cell $x$,
    \[
    f(1_x)=1_{f(x)}.
    \]
  \end{itemize}
  We denote by $\nMag$ the category of $n$-magmas and morphisms of $n$-magmas.
\end{paragr}
\begin{paragr}
  Let $n \in \nbar$. An \emph{$n$-category} $C$ is an $n$-magma such that the following axioms are satisfied:
  \begin{enumerate}
    \item for all $k,l \in \mathbb{N}$ with $k<l\leq n$, for all $k$-composable $l$-cells $x$ and $y$, we have 
    \[
    1_{x\underset{k}{\ast}y}=1_{x}\underset{k}{\ast}1_{y},
    \]
  \item for all $k,l \in \mathbb{N}$ with $k<l\leq n$, for all $l$-cells $x, y$ and $z$ such that $x$ and $y$ are $k$-composable, and $y$ and $z$ are $k$-composable, we have
    \[
    (x\comp_{k}y)\comp_{k}z=x\comp_k(y\comp_kz),
    \]
    \item for all $k, l \in \mathbb{N}$, for all $n$-cells $x,x',y$ and $y'$  such that
    \begin{itemize}
    \item[-] $x$ and $y$ are $l$-composable, $x'$ and $y'$ are $l$-composable,
    \item[-] $x$ and $x'$ are $k$-composable, $y$ and $y'$ are $k$-composable,
    \end{itemize}
    we have
    \[
    ((x \ast^n_k x')\ast^n_l (y \ast^n_k y'))=((x \ast^n_l y)\ast^n_k (x' \ast^n_l y')).
    \]
    \end{enumerate}
  \end{paragr}
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  \[
  \begin{tikzcd}[column sep=huge]
    n\Cat \ar[r,"\tau",""{name=B,above},""{name=C,below}] & (n\shortminus 1)\Cat \ar[l,bend right,"\iota"',""{name=A, below}] \ar[l,bend left,"\kappa",""{name=D, above}] \ar[from=A,to=B,symbol=\dashv]\ar[from=C,to=D,symbol=\dashv].
    \end{tikzcd}
  \]
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   For every $k\in \mathbb{N}$ such that $k < n$, we define the set $\Sigma^{+}\underset{C_k}{\times}\Sigma^{+}$ as the following fibred product
    \[
    \begin{tikzcd}
      \Sigma^{+}\underset{C_k}{\times}\Sigma^{+} \ar[r] \ar[dr,phantom,"\lrcorner", very near start] \ar[d] &\Sigma^{+} \ar[d,"t_k"]\\
      \Sigma^{+} \ar[r,"s_k"] & C_k.
      \end{tikzcd}
    \]
    That is, elements of $\Sigma^{+}\underset{C_k}{\times}\Sigma^{+}$ are pairs $(x,y)$ of well formed words such that $s_k(x)=t_k(y)$. We say that two well formed words $x$ and $y$ are \emph{$k$-composable} if the pair $(x,y)$ belongs to $\Sigma^{+}\times_{C_k}\Sigma^{+}$. 

\end{paragr}
We define $s_k , t_k: \Sigma^{+} \to C_k$ as iterated source and target (with $s_n=s$ and $t_n=t$ for consistency). We say that two well formed words $v$ and $w$ are \emph{parallel} if
\[s(v)=s(w) \text{ and }t(v)=t(w).\]
and we say that they are \emph{$k$-composable} for a $k\leq n$ if
\[s_k(v)=t_k(w).\]
 It is straightforward to check that
\[
\begin{tikzcd}
  C_0 & \ar[l,shift right,"t"'] \ar[l,shift left,"s"] \cdots & \ar[l,shift right,"t"'] \ar[l,shift left,"s"] C_{n \shortminus 1} & \ar[l,shift right,"t"'] \ar[l,shift left,"s"] \Sigma^{+}
  \end{tikzcd}
\]
is an $n$-graph and
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\section{Generating cells}
\begin{paragr}
  Let $n>0$, we define the category $n\CellExt$ of \emph{$n$-cellular extensions} as the following fibred product
  \begin{equation}\label{squarecellext}
    \begin{tikzcd}
  n\CellExt \ar[d] \ar[r] \ar[dr,phantom,"\lrcorner", very near start]&  (n \shortminus 1)\Cat \ar[d] \\
n\Grph  \ar[r,"\tau"] & (n \shortminus 1)\Grph,
  \end{tikzcd}
    \end{equation}
    where the right vertical arrow is the obvious forgetful functor.

    More concretely, an $n$-cellular extension can be encoded in the data of a quadruple $(\Sigma,C,s,t)$ where $\Sigma$ is a set, $C$ is a $(n\shortminus 1)$-category, $s$ and $t$ are maps
    \[
    s,t : \Sigma \to C_n
    \]
    such that
    \[
    \begin{tikzcd}
      C_{n-1} & \ar[l,shift right, "s"'] \ar[l,shift left,"t"] C_n & \ar[l,shift right, "s"'] \ar[l,shift left,"t"] \Sigma
      \end{tikzcd}
    \]
    satisfy the globular identities.

    Intuitively, a $n$-cellular extension is a $(n\shortminus 1)$-category with extra $n$-cells that make it a $n$-graph.  

    We will sometimes write
    \[
    \begin{tikzcd}
     C &\ar[l,shift right,"s"'] \ar[l,shift left,"t"]  \Sigma
      \end{tikzcd}
    \]
    to denote an $n$-cellular extension $(\Sigma,C,s,t)$. \remtt{Est-ce que je garde cette notation ?}
    
    A morphism of $n$-cellular extensions from  $(\Sigma,C,s,t)$ to $(\Sigma',C',s',t')$ consists of a pair $(\varphi,f)$ where $f : C \to C'$ is a morphism of $(n\shortminus 1)\Cat$ and $\varphi : \Sigma \to \Sigma'$ is a map such that the squares
    \[
    \begin{tikzcd}
      \Sigma \ar[r,"\varphi"] \ar[d,"s"] & \Sigma' \ar[d,"s'"] \\
      C_{n-1} \ar[r,"f_{n\shortminus 1}"] & C'_{n-1}
    \end{tikzcd}
    \text{ and }
        \begin{tikzcd}
      \Sigma \ar[r,"\varphi"] \ar[d,"t"] & \Sigma' \ar[d,"t'"] \\
      C_{n-1} \ar[r,"f_{n\shortminus 1}"] & C'_{n-1}
    \end{tikzcd}
        \]
        commute.
        
         Once again, we will use the notation $\tau$ for the functor
         \[
         \begin{aligned}
           \tau : n\CellExt &\to (n\shortminus 1)\Cat\\
           (\Sigma,C,s,t) &\mapsto C
           \end{aligned}
    \]
    which is simply the top horizontal arrow of square \eqref{squarecellext}. 
  \end{paragr}
\begin{paragr}
  Let $n>0$, we define the category $n\PCat$ of \emph{$n$-precategories} as the following fibred product
  \begin{equation}\label{squareprecat}
  \begin{tikzcd}
  n\PCat \ar[d] \ar[r] \ar[dr,phantom,"\lrcorner", very near start]& (n \shortminus 1)\Cat \ar[d] \\
  n\Mag \ar[r,"\tau"] & (n \shortminus 1)\Mag.
  \end{tikzcd}
  \end{equation}
  More concretely, objects of $n\PCat$ can be seen as $n$-magmas such that if we forget their $n$-cells then they satisfy the axioms of $(n\shortminus 1)$-category. The left vertical arrow of the previous square is easily seen to be full, and we will now consider that the category $n\PCat$ is a full subcategory of $n\Mag$. The top horizontal arrow of square \eqref{squareprecat} is simply the functor that forgets the $n$-cells. Once again, we will use the notation 
  \[
  \tau : n\PCat \to (n\shortminus 1)\Cat. 
  \]

  
  The commutative square
       \[
     \begin{tikzcd}
       n\Cat\ar[r,"\tau"] \ar[d] &(n \shortminus 1)\Cat\ar[d]\\
       n\Mag   \ar[r,"\tau"] & (n \shortminus 1)\Mag,
       \end{tikzcd}
     \]
     where the vertical arrows are the obvious forgetful functors, induces a canonical functor
     \[
     V : n\Cat \to n\PCat,
     \]
     which is also easily seen to be full. %and we will consider that $n\Cat$ is a full subcategory of $n\PCat$.

     Moreover, the canonical commutative diagram
     \[
     \begin{tikzcd}
       n\Mag \ar[d] \ar[r] & (n\shortminus 1)\Mag \ar[d] & (n\shortminus 1)\Cat \ar[d] \ar[l]\\
       n\Grph \ar[r] & (n\shortminus 1)\Grph & (n\shortminus 1)\Cat \ar[l]
     \end{tikzcd}
     \]
     induces a canonical functor
     \[
     W : n\PCat \to n\CellExt.
     \]
     For an $n$-precategory $C$, $W(C)$ is simply the cellular extension
     \[
         \begin{tikzcd}
     \tau(C) &\ar[l,shift right,"s"'] \ar[l,shift left,"t"]  C_n.
      \end{tikzcd}
     \]

     Finally, we define the functor
     \[
     U := W \circ V : n\Cat \to n\CellExt.
     \]
     The relation between $n\Cat$, $n\PCat$, $n\CellExt$ and $(n\shortminus 1)\Cat$ is summed up in the following commutative diagram:
     \[
     \begin{tikzcd}
       n\Cat \ar[rr,bend left,"U"]\ar[r,"V"] \ar[rrd,"\tau"'] & n\PCat \ar[r,"W"]\ar[rd,"\tau"] & n\CellExt \ar[d,"\tau"] \\
       &&(n\shortminus 1)\Cat.
       \end{tikzcd}
     \]
     We will now explicitely construct a left adjoint of $U$. In order to do that, we will successively construct left adjoints of $W$ and $V$.
\end{paragr}
\begin{paragr}
Let $E=(\Sigma,C,s,t)$ be an $n$-cellular extension. We define an $n$-precategory $W_!(E)$ with
\begin{itemize}
\item[-] $\tau(W_!(E))=C$,
\item[-] $W_!(E)_n=\Sigma^{+}$,
\item[-] source and target maps $\Sigma^+ \to C_{n-1}$ as defined in the previous paragraph,
\item[-] for every $x \in C_{n-1}$,
  \[1_x := (\ii_x)\]
\item[-] for every $v,w \in \Sigma^+$ that are $k$-composable for a $k<n$,
  \[
  v\comp_kw := (v \fcomp_k w).
  \]
\end{itemize}
It is straightforward to check that this defines an $n$-precategory. Let $(\varphi,f) : E \to E'$ be a morphism of $n$-cellular extensions. We define a morphism of $n$-precategories $W_!(\varphi,f) : W_!(E) \to W_!(E')$ with
\end{paragr}
 For $u : A \to B$ in $\CCat$, let
  \[
  u^* : \C(A) \to \C(B)
  \]
  be the functor induced by post-composition. For $\begin{tikzcd}        \sD(B)\ar[r,bend left,"u^*",""{name=U,below}] \ar[r,bend right,"v^*"',""{name=D,above}] & \sD(A) \ar[from=U,to=D,Rightarrow,"\alpha^*"] \end{tikzcd}$
  
 Note that we have a canonical isomorphism
  \[
  \C(e) \simeq \C
  \]
  and for any small category $A$, the functor
  \[
  p_A^* : \C(e) \to \C(A)
  \]
  is canonically isomorphic with the diagonal functor $\Delta : \C \to \C(A)$ that sends an object $X$ of $\C$ to the constant diagram $A \to \C$ with value $X$.
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\begin{paragr}
  Let $(\C,\W)$ be a localizer and $A$ a small category. We denote by $\C^A$ the category of functors from $A$ to $\C$ and natural transformations between them. An arrow $\alpha : d \to d'$ of $\C^A$ is a \emph{pointwise weak equivalences} when $\alpha_a : d(a) \to d'(a)$ belongs to $\W$ for every $a \in A$. We denote by $\W_A$ the class of pointwise weak equivalences. This defines a localizer $(\C^A,\W_A)$.

  For every $u : A \to B$ morphism of $\Cat$, the functor induced by pre-composition
  \[
  u^* : \C^B \to \C^A
  \]
  preserves pointwise weak equivalences. Hence, there is an induced functor between the localized categories still denoted by $u^*$:
  \[
  u^* : \Ho(\C^B) \to \Ho(\C^A).
  \]
\end{paragr}
  Dually, if we are given a square in $\CCat$ of the form
   \[
     \begin{tikzcd}
    A \ar[r,"f"] \ar[d,"u"'] & B \ar[d,"v"]\\
    C \ar[r,"g"'] & D \ar[from=2-1,to=1-2,Rightarrow,"\alpha"]
  \end{tikzcd}
     \]
     and if $\sD$ has right Kan extension, then we obtain the cohomological base change morphism induced by $\alpha$:
     \[
     
     \]
  \end{paragr}
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%%% Définition transfo pseudo-naturelle

   \iffalse
    such that the following axioms are satisfied:
    \begin{itemize}[label=-]
    \item for every small category $A$,
      \[
      F_{1_A}=1_{F(A)},
      \]
    \item for every pair of composable arrows $A \overset{u}{\rightarrow} B \overset{v}{\rightarrow} C$ in $\CCat$,
      \[
      \begin{tikzcd}
        \sD(C) \ar[d,"v^*"'] \ar[r,"F_C"] & \sD'(C) \ar[d,"v^*"]\\
        \ar[from=1-2,to=2-1,Rightarrow,"F_v","\sim"']
        \sD(B) \ar[d,"u^*"'] \ar[r,"F_B"] & \sD'(B) \ar[d,"u^*"]\\
        \sD(A) \ar[r,"F_A"'] & \sD'(A)
        \ar[from=2-2,to=3-1,Rightarrow,"F_u","\sim"']
      \end{tikzcd}
      =
      \begin{tikzcd}
        \sD(C) \ar[dd,"(vu)^*"'] \ar[r,"F_C"] & \sD'(C) \ar[dd,"(vu)^*"]\\
        &\\
        \sD(A) \ar[r,"F_A"'] & \sD'(A),
        \ar[from=1-2,to=3-1,Rightarrow,"F_{vu}","\sim"']
        \end{tikzcd}
      \]
    \item for every $\begin{tikzcd} A \ar[r,bend left,"u",""{name=A,below}] \ar[r,bend right,"v"',""{name=B,above}] & B  \ar[from=A,to=B,Rightarrow,"\alpha"] \end{tikzcd}$ in $\CCat$,
      \[
      \begin{tikzcd}
        \sD(B) \ar[d,"u^*",""{name=A,below}] \ar[r,"F_B"] & \sD'(B) \ar[d,"u^*"]\\
        \sD(A) \ar[r,"F_A"'] & \sD'(A)
        \ar[from=1-2,to=2-1,Rightarrow,"F_u","\sim"']
        \ar[from=1-1,to=2-1,bend right=55,"v^*"',""{name=B,below}]
        \ar[from=A,to=B,Rightarrow,"\alpha^*"',near start]
      \end{tikzcd}
      =
            \begin{tikzcd}
        \sD(B) \ar[d,"v^*"] \ar[r,"F_B"] & \sD'(B) \ar[d,"v^*"',""{name=D,below}]\\
        \sD(A) \ar[r,"F_A"'] & \sD'(A).
        \ar[from=1-2,to=2-1,Rightarrow,"F_v","\sim"']
        \ar[from=1-2,to=2-2,bend left=55,"u^*",""{name=C,above}]
        \ar[from=C,to=D,Rightarrow,"\alpha^*"',near start]
        \end{tikzcd}
      \]
    \end{itemize}
    \remtt{Ai-je vraiment besoin de donner la définition de transfo pseudo-naturelle ? }\fi

    %%% Localization derivator

    It is straightforward to check that we have the following universal property: for any op-prederivator $\sD$, the functor induced by pre-composition
    \[
    \gamma^* : \underline{\Hom}(\sD_{(\C,\W)},\sD) \to \underline{\Hom}(\C,\sD)
    \]
    is fully faithful and its essential image consists of morphisms of op-prederivators $F : \C \to \sD$ such that for every small category $A$, $F_A : \C(A) \to \sD(A)$ sends morphisms of $\W_A$ to isomorphisms of $\sD(A)$.

    This universal property is a higher version of the universal property of localization seen in \ref{paragr:loc}. It follows that given two localizers $(\C,\W)$ and $(\C',\W')$ and a functor $F : \C \to \C'$, if $F$ preserves weak equivalences, then there exists a morphism of op-prederivators $ \overline{F} : \sD_{(\C,\W)} \to \sD_{(\C',\W')}$ such that the square 
    \[
    \begin{tikzcd}
    \C \ar[r,"F"] \ar[d,"\gamma"] & \C' \ar[d,"\gamma'"]\\
    \sD_{(\C,\W)} \ar[r,"\overline{F}"] & \sD_{(\C',\W')}
  \end{tikzcd}
    \]
    is commutative.

    %%% Left derived functor derivator

      \begin{definition}
    Let $(\C,\W)$ and $(\C',\W')$ be two localizers. A functor $F : \C \to \C'$ is \emph{left derivable in the sense of derivators} if there exists a morphism $\LL F : \sD_{(\C,\W)} \to \sD_{(\C',\W')}$ and a $2$-morphism
  \[
  \begin{tikzcd}
    \C \ar[r,"F"] \ar[d,"\gamma"] & \C' \ar[d,"\gamma'"]\\
    \sD_{(\C,\W)} \ar[r,"\LL F"'] & \sD_{(\C',\W')}
    \arrow[from=2-1, to=1-2,"\alpha",Rightarrow]
  \end{tikzcd}
  \]
    that makes $\LL F$ the \emph{right} Kan extension of $\gamma' \circ F$ along $\gamma$ in the $2$-category of op-prederivators:
    \end{definition}
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      %% Old version of representable op-prederivator

      \iffalse
\begin{example}\label{ex:repder}
  Let $\C$ be a category. For $A$ a small category, let $\C(A)$ be the category of functors $A \to \C$ and natural transformations between them.
  This canonically defines an op-prederivator
  \begin{align*}
    \C : \CCat^{op} &\to \CCAT \\
    A &\mapsto \C(A)
  \end{align*}
  where for any $u : A \to B$ in $\CCat$, the functor
  \[
  u^* : \C(A) \to \C(B)
  \]
  is induced from $u$ by pre-composition, and similarly for $\begin{tikzcd}A \ar[r,bend left,"u",""{name=A,below}] \ar[r,bend right, "v"',""{name=B,above}] & B \ar[from=A,to=B,Rightarrow,"\alpha"]\end{tikzcd}$ in $\CCat$, the natural transformation
  \[
  \begin{tikzcd}
    \C(B) \ar[r,bend left,"u^*",""{name=A,below}] \ar[r,bend right, "v^*"',""{name=B,above}] & \C(A) \ar[from=A,to=B,Rightarrow,"\alpha^*"]
    \end{tikzcd}
  \]
  is induced by pre-composition. This op-prederivator is sometimes referred to as the op-prederivator \emph{represented} by $\C$. Notice that for the terminal category $e$, we have a canonical isomorphism
  \[
  \C(e) \simeq \C
  \]
  that we shall use without further reference.
\end{example}\fi