The Stacks project

14.4 Simplicial objects as presheaves

Another observation is that we may think of a simplicial object of $\mathcal{C}$ as a presheaf with values in $\mathcal{C}$ over $\Delta$. See Sites, Definition 7.2.2. And in fact, if $U$, $U'$ are simplicial objects of $\mathcal{C}$, then we have

Some of the material below could be replaced by the more general constructions in the chapter on sites. However, it seems a clearer picture arises from the arguments specific to simplicial objects.