is homotopy cocartesian. But, since $L \simeq c\circ i_!$, it follows from Lemma \ref{cor:hmtpysquaregraph} that this last square is weakly equivalent to the square of simplicial sets

This square is cocartesian because $i_!$ is a left adjoint and since $i_!$ preserves monomorphism (Lemma \ref{lemma:monopreserved}), the result follows from the fact that monomorphisms are cofibrations of simplicial sets for the standard Quillen model structure and \todo{ref}.

\end{proof}

Actually, by working a little more, we obtain the slightly more general result below.

\begin{proposition}

Let

\[

\begin{tikzcd}

A \ar[d,"\alpha"]\ar[r,"\beta"] B \ar[d,"\delta"]\\

C \ar[r,"\gamma"] D

\end{tikzcd}

\]

be a cocartesian square in $\RGrph$. Suppose that both following conditions are satisfied

\begin{itemize}[label=\alph*)]

\item Either $\alpha$ or $\beta$ is injective on objects.

\item Either $\alpha$ or $\beta$ is injective on morphisms.

is a \emph{homotopy} cocartesian square of $\Cat$ equipped with the Thomason weak equivalences.

\end{proposition}

\begin{proof}

The case where $\alpha$ or $\beta$ is injective both on objects and morphisms is Corollary \ref{cor:hmtpysquaregraph} hence we only have to treat the case when $\alpha$ is injective on objects and $\beta$ is injective on arrows. Let use denote by $E$ the set of objects of $C$ that lies in the image of $\alpha$. For each element $x$ of $E$, we denote by

\end{proof}

\begin{example}[Adding a generator]

Let $C$ be a free category, $A$ and $B$ (possibly equal) two objects of $C$ and let $C'$ be the category obtained from $C$ by adding a generator $A \to B$, i.e. defined with the following cocartesian square: