Remark 7.13.6. (Skip on first reading.) Let \mathcal{C} and \mathcal{D} be sites. Let us use the definition of tautologically equivalent families of maps, see Definition 7.8.2 to (slightly) weaken the conditions defining continuity. Let u : \mathcal{C} \to \mathcal{D} be a functor. Let us call u quasi-continuous if for every \mathcal{V} = \{ V_ i \to V\} _{i\in I} \in \text{Cov}(\mathcal{C}) we have the following
the family of maps \{ u(V_ i) \to u(V)\} _{i\in I} is tautologically equivalent to an element of \text{Cov}(\mathcal{D}), and
for any morphism T \to V in \mathcal{C} the morphism u(T \times _ V V_ i) \to u(T) \times _{u(V)} u(V_ i) is an isomorphism.
We are going to see that Lemmas 7.13.2 and 7.13.3 hold in case u is quasi-continuous as well.
We first remark that the morphisms u(V_ i) \to u(V) are representable, since they are isomorphic to representable morphisms (by the first condition). In particular, the family u(\mathcal{V}) = \{ u(V_ i) \to u(V)\} _{i\in I} gives rise to a zeroth Čech cohomology group H^0(u(\mathcal{V}), \mathcal{F}) for any presheaf \mathcal{F} on \mathcal{D}. Let \mathcal{U} = \{ U_ j \to u(V)\} _{j \in J} be an element of \text{Cov}(\mathcal{D}) tautologically equivalent to \{ u(V_ i) \to u(V)\} _{i \in I}. Note that u(\mathcal{V}) is a refinement of \mathcal{U} and vice versa. Hence by Remark 7.10.7 we see that H^0(u(\mathcal{V}), \mathcal{F}) = H^0(\mathcal{U}, \mathcal{F}). In particular, if \mathcal{F} is a sheaf, then \mathcal{F}(u(V)) = H^0(u(\mathcal{V}), \mathcal{F}) because of the sheaf property expressed in terms of zeroth Čech cohomology groups. We conclude that u^ p\mathcal{F} is a sheaf if \mathcal{F} is a sheaf, since H^0(\mathcal{V}, u^ p\mathcal{F}) = H^0(u(\mathcal{V}), \mathcal{F}) which we just observed is equal to \mathcal{F}(u(V)) = u^ p\mathcal{F}(V). Thus Lemma 7.13.2 holds. Lemma 7.13.3 follows immediately.
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