Given a category $\mathcal{C}$ with a notion of covering $\{ U_{i} \rightarrow X \}$ for an object $X$ (say $\mathcal{C}$ is a Grothendieck site), we can form the Cech nerve
$$ \cdots \coprod_{i}{U_{ijk}} \substack{\rightarrow \\ \rightarrow \\ \rightarrow}\coprod_{i}{U_{ij}} \rightrightarrows \coprod_{i} U_{i}$$
(In the notation, I've suppressed degeneracy maps going from right to left.)
This can be viewed in two ways. 1. As a simplicial object in simplicial presheaves, by considering each $U_{i}$ as a simplicial presheaf constant in the simplicial direction. I'll denote that by $U_{\bullet}$. 2. As a simplicial object in presheaves and hence a simplicial presheaf. I'll denote that by $\check{U}_{\bullet}$.
The latter $\check{U}_{\bullet}$ can be shown to be level-wise weakly equivalent to $\operatorname{colim}(\coprod_{i}{U_{ij}} \rightrightarrows \coprod_{i} U_{i})$, where we consider this as a simplicial presheaf constant in the simplicial direction.
On the other hand, we could compute $\operatorname{hocolim}(U_{\bullet})$, and from things I read, this is supposed to be identified with/weakly equivalent to $\check{U}_{\bullet}$. One reason I am having difficulty seeing this is that I don't really understand $\operatorname{hocolim}(U_{\bullet})$. Since each object in $U_{\bullet}$ is cofibrant, I would guess I could just take the usual colimit, but this seems to produce something constant in the simplicial direction, which is clearly wrong.
So the question is:
How to see if there is a weak equivalence $\operatorname{hocolim}(U_{\bullet}) \rightarrow \check{U}_{\bullet}$?
Probably if I read through the many pages of material suggested in the comments to this question, I'd be able to figure this out. But a more direct answer would make that reading more fruitful for me, I think. At least, pointing out what things I need to know to figure this out would help.
