The Stacks project

Remark 85.5.1. Let $\mathcal{C}_ n, f_\varphi , u_\varphi $ and $\mathcal{C}'_ n, f'_\varphi , u'_\varphi $ be as in Situation 85.3.3. A morphism $h$ between simplicial sites will mean

  1. Morphisms of sites $h_ n : \mathcal{C}_ n \to \mathcal{C}'_ n$ such that $f'_\varphi \circ h_ n = h_ m \circ f_\varphi $ as morphisms of sites for all $\varphi : [m] \to [n]$.

  2. Cocontinuous functors $v_ n : \mathcal{C}_ n \to \mathcal{C}'_ n$ inducing morphisms of topoi $h_ n : \mathop{\mathit{Sh}}\nolimits (\mathcal{C}_ n) \to \mathop{\mathit{Sh}}\nolimits (\mathcal{C}'_ n)$ such that $u'_\varphi \circ v_ n = v_ m \circ u_\varphi $ as functors for all $\varphi : [m] \to [n]$.

In both cases we have $f'_\varphi \circ h_ n = h_ m \circ f_\varphi $ as morphisms of topoi, see Sites, Lemma 7.21.2 for case B and Sites, Definition 7.14.5 for case A.

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