The Stacks project

The categories $\mathcal I_{V_j}^u$ may be empty. This happens for example if $\mathcal C = \varnothing$ is the empty site, but it could also just happen that none of the $V_j$ admit a map to any $u(U)$.

Dear Remy, I think the lemma (and the proof) is OK. The key is that the definition of a "morphism of sites" forces exactness of the functor $f^{-1} = u_s$. For example, if $\mathcal{C}$ is the empty site, then $\textit{Sh}(\mathcal{D})$ has to be the empty topos (because the final and initial object of it have to be the same as the pullback of such on $\mathcal{C}$ by an exact functor). And this in turn means that every object $V$ of $\mathcal{D}$ has an EMPTY covering! Thus the lemma holds in this case. OK?

You're right; I just realised the same thing.