Suppose that I have a diagram of simplicial sets $X_\bullet:\mathscr{C} \to Set^{\Delta^{op}},$ with $\mathscr{C}$ a small category such that for each $C \in \mathscr{C},$ $X_\bullet(C)$ is a Kan complex. Let $X_0$ denote the diagram of $0$-simplices, but regarded again as a simplicial diagram on $\mathscr{C}$ (which is simplicially constant). There is a canonical natural transformation $X_0 \to X_\bullet.$

**Question:** Under what conditions will the induced map $$\mathbf{hocolim} X_0 \to \mathbf{hocolim} X_\bullet$$ between the homotopy colimits (computed in Quillen model structure on simplicial sets) be a weak equivalence?

To be clear, I'm not interested in pathalogical conditions like $X_\bullet$ is itself simplicially constant. I have a very specific example in mind which is not of this trivial form.

Here is a non-trivial example where it does work:

**Example:**

Take $\mathbf{M}$ to be a (almost) simplicial model category in which every object is cofibrant, and suppose that there is a cosimplicial object $I^\bullet$ such that $I^0$ is terminal and the simplicial enrichment is given by $$Map(C,D)_n=Hom(C \times I^n,D).$$ (For example, take the opposite category of the projective model structure on commutative dg-algebras). Let $D$ and $E$ be fibrant. Consider the category $\mathscr{C}$ to be the (opposite of the) category of trivial fibrations over $D$ with the arrows being commutative triangles. Let $X_\bullet$ be defined by

$$X_\bullet(\varphi:D' \to D)=Map(D',E)_\bullet.$$ Then both homotopy colimits are equivalent to the mapping space $Map(D,E).$