Simplicial presheaves that are colimits of themselves?

Question

Suppose $C$ is a small category and $X_{\bullet}$ is a simplicial object in $C$. In particular, by composing with Yoneda $$y:C \to Set^{C^{op}}$$ $y(X)_{\bullet}$ is a simplicial presheaf. I believe it is true that in (either the injective or projective) model structure on simplicial presheaves, that $y(X)_{\bullet}$ "is the homotopy colimit of itself", that is, the homotopy colimit of the functor $$\Delta^{op} \stackrel{X}{\longrightarrow} C \stackrel{y}{\longrightarrow} Set^{C^{op}} \to \left(Set^{\Delta^{op}}\right)^{C^{op}} $$ is the simplicial presheaf $y(X)_{\bullet}$ itself. Does someone know a quick proof. I think it should be simple, I just cannot think very well at the moment. I'm also fine with an infinity category answer to this.

Answer 1

Take the diagonal of $\Delta^{op} \to \left(Set^{\Delta^{op}}\right)^{C^{op}}$. If I understand the rightmost arrow in your question correctly, then this is just $y(X_\bullet)$.

Answer 2

Yes, that's right. I once wrote out some details behind this (of course it's standard) in the nLab at

• Reedy model structure -- Examples -- Over the simplex category

and at

• Homotopy limit -- Examples -- Homotopy colimits over simplicial diagrams .