Denote $\newcommand{\simpcat}{\boldsymbol\Delta} \simpcat$ the simplex category (category of finite ordinals with non decreasing maps). The diagonal functor $\delta \colon \simpcat \to \simpcat\times\simpcat$ induces a functor $$ \newcommand{\psh}{\widehat} \delta^\ast \colon \psh{\simpcat \times \simpcat} \to \psh\simpcat $$ mapping a bisimplicial set $K$ to the simplicial set $\newcommand{\ordinal}[1]{[#1]} \ordinal n \mapsto K_{n,n}$.
Because $\newcommand{\Set}{\mathsf{Set}}\Set$ is complete, $\delta^\ast$ has a right adjoint, namely the right Kan extension functor along $\newcommand{\op}[1]{#1^\text{op}}\op\delta$. Now, the paper I'm reading states that this right adjoint $\delta_\ast$ maps any simplicial set $X$ to the bisimplicial set $$ (\ordinal m ,\ordinal n) \mapsto \hom_{\psh{\simpcat}}(\Delta[m]\times\Delta[n],X) $$ where $\Delta[-]$ is the Yoneda embedding $\simpcat \hookrightarrow \psh{\simpcat}$.
I'm having trouble deriving this explicit form from the general form of right Kan extensions. Here is what I've done so far : $$ \begin{aligned} (\delta_\ast X)_{m,n} &= \lim_{\left( (\ordinal m, \ordinal n) \downarrow\op\delta \right)} X \\ &= \lim_{\left( (\ordinal m, \ordinal n) \downarrow\op\delta \right)} \hom_{\psh{\simpcat}}(\Delta[-],X) \\ &= \hom_{\psh{\simpcat}}\left(\operatorname*{colim}_{\left( (\ordinal m, \ordinal n) \downarrow\op\delta \right)}\Delta[-],X\right) \\ \end{aligned} $$ But I can't manage to show that $\operatorname*{colim}_{\left( (\ordinal m, \ordinal n) \downarrow\op\delta \right)}\Delta[-]$ is actually $\Delta[m] \times \Delta[n]$. The only cocone over the diagram $\Delta[-] \colon \left( (\ordinal m, \ordinal n) \downarrow\op\delta \right) \to \psh{\simpcat}$ I can imagine is the tautological one : for any object $(f,g)$ of $\left( (\ordinal m, \ordinal n) \downarrow\op\delta \right)$ (with $f \colon \ordinal k \to \ordinal m$ and $g \colon \ordinal k \to \ordinal n$), denote $\mu_{(f,g)}$ the morphism $\Delta[k] \to \Delta[m] \times \Delta[n]$ corresponding under Yoneda's lemma to the $k$-simplices $(f,g)$ of $\Delta[m] \times \Delta[n]$. But I don't see why this cocone is universal (actually, I'm not even sure it is a cocone...).
Any hint ?