Example 7.22.3. This example continues the discussion of Example 7.14.3 from which we borrow the notation $\mathcal{C}, \tau , \tau ', \epsilon$. Observe that the identity functor $v : \mathcal{C}_{\tau '} \to \mathcal{C}_\tau$ is a continuous functor and the identity functor $u : \mathcal{C}_\tau \to \mathcal{C}_{\tau '}$ is a cocontinuous functor. Moreover $u$ is left adjoint to $v$. Hence the results of Lemmas 7.22.1 and 7.22.2 apply and we conclude $v$ defines a morphism of sites, namely

$\epsilon : \mathcal{C}_\tau \longrightarrow \mathcal{C}_{\tau '}$

whose corresponding morphism of topoi is the same as the morphism of topoi associated to the cocontinuous functor $u$.

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