@@ -336,3 +336,27 @@ It is straightforward to check that this defines an $n$-precategory. Let $(\varp
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 $X$.
\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 $\alpha$:
