The Stacks project

Proof. Case A. Let $u_ n : \mathcal{D} \to \mathcal{C}_ n$ be the common value of the functors $u_\varphi \circ u_0$ for $\varphi : [0] \to [n]$. Then $u_ n$ corresponds to a morphism of sites $a_ n : \mathcal{C}_ n \to \mathcal{D}$, see Sites, Lemma 7.14.4. The same lemma shows that for all $\varphi : [m] \to [n]$ we have $a_ m \circ f_\varphi = a_ n$.

Case B. Let $u_ n : \mathcal{C}_ n \to \mathcal{D}$ be the common value of the functors $u_0 \circ u_\varphi$ for $\varphi : [0] \to [n]$. Then $u_ n$ is cocontinuous and hence defines a morphism of topoi $a_ n : \mathop{\mathit{Sh}}\nolimits (\mathcal{C}_ n) \to \mathop{\mathit{Sh}}\nolimits (\mathcal{D)}$, see Sites, Lemma 7.21.2. The same lemma shows that for all $\varphi : [m] \to [n]$ we have $a_ m \circ f_\varphi = a_ n$.

Consider the functor $a^{-1} : \mathop{\mathit{Sh}}\nolimits (\mathcal{D}) \to \mathop{\mathit{Sh}}\nolimits (\mathcal{C}_{total})$ which to a sheaf of sets $\mathcal{G}$ associates the sheaf $\mathcal{F} = a^{-1}\mathcal{G}$ whose components are $a_ n^{-1}\mathcal{G}$ and whose transition maps $\mathcal{F}(\varphi )$ are the identifications

for $\varphi : [m] \to [n]$, see the description of $\mathop{\mathit{Sh}}\nolimits (\mathcal{C}_{total})$ in Lemma 85.3.4. Since the functors $a_ n^{-1}$ are exact, $a^{-1}$ is an exact functor. Finally, for $a_* : \mathop{\mathit{Sh}}\nolimits (\mathcal{C}_{total}) \to \mathop{\mathit{Sh}}\nolimits (\mathcal{D})$ we take the functor which to a sheaf $\mathcal{F}$ on $\mathop{\mathit{Sh}}\nolimits (\mathcal{D})$ associates

Here the two maps come from the two maps $\varphi : [0] \to [1]$ via

where the first arrow comes from $1 \to f_{\varphi , *} f_\varphi ^{-1}$. Let $\mathcal{G}_\bullet$ denote the constant coosimplicial sheaf with value $\mathcal{G}$ and let $a_{\bullet , *}\mathcal{F}$ denote the cosimplicial sheaf having $a_{n, *}\mathcal{F}_ n$ in degree $n$. By the usual adjuntion for the morphisms of topoi $a_ n$ we see that a map $a^{-1}\mathcal{G} \to \mathcal{F}$ is the same thing as a map

of cosimplicial sheaves. By the dual to Simplicial, Lemma 14.20.2 this is the same thing as a map $\mathcal{G} \to a_*\mathcal{F}$. Thus $a^{-1}$ and $a_*$ are adjoint functors and we obtain our morphism of topoi $a$¹. The equalities $a \circ g_ n = f_ n$ follow immediately from the definitions. $\square$