The Stacks project

Remark 85.5.4. Let $\mathcal{C}_ n, f_\varphi , u_\varphi $ and $\mathcal{C}'_ n, f'_\varphi , u'_\varphi $ be as in Situation 85.3.3. Let $a_0$, resp. $a'_0$ be an augmentation towards a site $\mathcal{D}$, resp. $\mathcal{D}'$ as in Remark 85.4.1. Let $h$ be a morphism between simplicial sites as in Remark 85.5.1. We say a morphism of topoi $h_{-1} : \mathop{\mathit{Sh}}\nolimits (\mathcal{D}) \to \mathop{\mathit{Sh}}\nolimits (\mathcal{D}')$ is compatible with $h$, $a_0$, $a'_0$ if

1. $h_{-1}$ comes from a morphism of sites $h_{-1} : \mathcal{D} \to \mathcal{D}'$ such that $a'_0 \circ h_0 = h_{-1} \circ a_0$ as morphisms of sites.

2. $h_{-1}$ comes from a cocontinuous functor $v_{-1} : \mathcal{D} \to \mathcal{D}'$ such that $u'_0 \circ v_0 = v_{-1} \circ u_0$ as functors.

In both cases we have $a'_0 \circ h_0 = h_{-1} \circ a_0$ as morphisms of topoi, see Sites, Lemma 7.21.2 for case B and Sites, Definition 7.14.5 for case A.