@@ -319,18 +319,24 @@ We now apply Corollary \ref{cor:hmtpysquaregraph} and Proposition \ref{prop:hmtp

\[

X : \Delta^{op}\times\Delta^{op}\to\Set.

\]

In a similar fashion as for simplicial sets (\ref{paragr:simpset}), for $n,m \geq0$, we use the notation

In a similar fashion as for simplicial sets (\ref{paragr:simpset}), for $n,m \geq0$, we use the following notations for the face and degeneracy operators:

Recall now from \cite[Proposition 1.2]{moerdijk1989bisimplicial} that the category of bisimplicial sets can be equipped with a model structure whose weak equivalences are diagonal weak equivalences and cofibrations are monomorphisms. We shall refer to this model structure as the \emph{diagonal model structure}. Since $\delta^*$ preserves monomorphisms (and diagonal weak equivalences trivially), it is left Quillen. In fact, the following proposition tells us more. We denote by $\delta_*$ the right adjoint of $\delta^*$.

Recall from \cite[Proposition 1.2]{moerdijk1989bisimplicial} that the category of bisimplicial sets can be equipped with a model structure whose weak equivalences are diagonal weak equivalences and whose cofibrations are monomorphisms. We shall refer to this model structure as the \emph{diagonal model structure}. Since $\delta^*$ preserves monomorphisms (and diagonal weak equivalences trivially), it is left Quillen. In fact, the following proposition tells us more. We denote by $\delta_*$ the right adjoint of $\delta^*$.

\end{paragr}

\begin{proposition}

Consider that $\Psh{\Delta\times\Delta}$ is equipped with the diagonal model structure. Then, the adjunction