The Stacks project

Proof. Case A. Say $h_ n$ corresponds to the continuous functor $v_ n : \mathcal{C}'_ n \to \mathcal{C}_ n$. Then we can define a functor $v_{total} : \mathcal{C}'_{total} \to \mathcal{C}_{total}$ by using $v_ n$ in degree $n$. This is clearly a continuous functor (see definition of coverings in Lemma 85.3.1). Let $h_{total}^{-1} = v_{total, s} : \mathop{\mathit{Sh}}\nolimits (\mathcal{C}'_{total}) \to \mathop{\mathit{Sh}}\nolimits (\mathcal{C}_{total})$ and $h_{total, *} = v_{total}^ s = v_{total}^ p : \mathop{\mathit{Sh}}\nolimits (\mathcal{C}_{total}) \to \mathop{\mathit{Sh}}\nolimits (\mathcal{C}'_{total})$ be the adjoint pair of functors constructed and studied in Sites, Sections 7.13 and 7.14. To see that $h_{total}$ is a morphism of topoi we still have to verify that $h_{total}^{-1}$ is exact. We first observe that $(g'_ n)^{-1} \circ h_{total, *} = h_{n, *} \circ g_ n^{-1}$; this is immediate by computing sections over an object $U$ of $\mathcal{C}'_ n$. Thus, if we think of a sheaf $\mathcal{F}$ on $\mathcal{C}_{total}$ as a system $(\mathcal{F}_ n, \mathcal{F}(\varphi ))$ as in Lemma 85.3.4, then $h_{total, *}\mathcal{F}$ corresponds to the system $(h_{n, *}\mathcal{F}_ n, h_{n, *}\mathcal{F}(\varphi ))$. Clearly, the functor $(\mathcal{F}'_ n, \mathcal{F}'(\varphi )) \to (h_ n^{-1}\mathcal{F}'_ n, h_ n^{-1}\mathcal{F}'(\varphi ))$ is its left adjoint. By uniqueness of adjoints, we conclude that $h_{total}^{-1}$ is given by this rule on systems. In particular, $h_{total}^{-1}$ is exact (by the description of sheaves on $\mathcal{C}_{total}$ given in the lemma and the exactness of the functors $h_ n^{-1}$) and we have our morphism of topoi. Finally, we obtain $g_ n^{-1} \circ h_{total}^{-1} = h_ n^{-1} \circ (g'_ n)^{-1}$ as well, which proves that the displayed diagram of the lemma commutes.

Case B. Here we have a functor $v_{total} : \mathcal{C}_{total} \to \mathcal{C}'_{total}$ by using $v_ n$ in degree $n$. This is clearly a cocontinuous functor (see definition of coverings in Lemma 85.3.2). Let $h_{total}$ be the morphism of topoi associated to $v_{total}$. The commutativity of the displayed diagram of the lemma follows immediately from Sites, Lemma 7.21.2. Taking left adjoints the final equality of the lemma becomes

This follows immediately from the explicit description of the functors $(g'_ n)^{Sh}_!$ and $g^{Sh}_{n!}$ in Lemma 85.3.5, the fact that $h_ n^{-1} \circ (f'_\varphi )^{-1} = f_\varphi ^{-1} \circ h_ m^{-1}$ for $\varphi : [m] \to [n]$, and the fact that we already know $h_{total}^{-1}$ commutes with restrictions to the degree $n$ parts of the simplicial sites. $\square$