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The Stacks project

Proposition 7.14.7. Let \mathcal{C} and \mathcal{D} be sites. Let u : \mathcal{C} \to \mathcal{D} be continuous. Assume furthermore the following:

  1. the category \mathcal{C} has a final object X and u(X) is a final object of \mathcal{D} , and

  2. the category \mathcal{C} has fibre products and u commutes with them.

Then u defines a morphism of sites \mathcal{D} \to \mathcal{C}, in other words u_ s is exact.

Proof. This follows from Lemmas 7.5.2 and 7.14.6. \square


Comments (1)

Comment #993 by on

Suggested slogan: Continuous functors preserving final objects and fibre products induce morphisms of sites.

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