The Stacks project

Example 7.14.3. Let $\mathcal{C}$ be a category. Let

\[ \text{Cov}(\mathcal{C}) \supset \text{Cov}'(\mathcal{C}) \]

be two sets of families of morphisms with fixed target which turn $\mathcal{C}$ into a site. Denote $\mathcal{C}_\tau $ the site corresponding to $\text{Cov}(\mathcal{C})$ and $\mathcal{C}_{\tau '}$ the site corresponding to $\text{Cov}'(\mathcal{C})$. We claim the identity functor on $\mathcal{C}$ defines a morphism of sites

\[ \epsilon : \mathcal{C}_\tau \longrightarrow \mathcal{C}_{\tau '} \]

Namely, observe that $\text{id} : \mathcal{C}_{\tau '} \to \mathcal{C}_\tau $ is continuous as every $\tau '$-covering is a $\tau $-covering. Thus the functor $\epsilon _* = \text{id}^ s$ is the identity functor on underlying presheaves. Hence the left adjoint $\epsilon ^{-1}$ of $\epsilon _*$ sends a $\tau '$-sheaf $\mathcal{F}$ to the $\tau $-sheafification of $\mathcal{F}$ (by the universal property of sheafification). Finite limits of $\tau '$-sheaves agree with finite limits of presheaves (Lemma 7.10.1) and $\tau $-sheafification commutes with finite limits (Lemma 7.10.14). Thus $\epsilon ^{-1}$ is left exact. Since $\epsilon ^{-1}$ is a left adjoint it is also right exact (Categories, Lemma 4.24.6). Thus $\epsilon ^{-1}$ is exact and we have checked all the conditions of Definition 7.14.1.

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