Lemma 7.8.7. Let $\mathcal{C}$ be a category. Let $\text{Cov}_ i$, $i = 1, 2$ be two sets of families of morphisms with fixed target which each define the structure of a site on $\mathcal{C}$.

If every $\mathcal{U} \in \text{Cov}_1$ is tautologically equivalent to some $\mathcal{V} \in \text{Cov}_2$, then $\mathop{\mathit{Sh}}\nolimits (\mathcal{C}, \text{Cov}_2) \subset \mathop{\mathit{Sh}}\nolimits (\mathcal{C}, \text{Cov}_1)$. If also, every $\mathcal{U} \in \text{Cov}_2$ is tautologically equivalent to some $\mathcal{V} \in \text{Cov}_1$ then the category of sheaves are equal.

Suppose that for each $\mathcal{U} \in \text{Cov}_1$ there exists a $\mathcal{V} \in \text{Cov}_2$ such that $\mathcal{V}$ refines $\mathcal{U}$. In this case $\mathop{\mathit{Sh}}\nolimits (\mathcal{C}, \text{Cov}_2) \subset \mathop{\mathit{Sh}}\nolimits (\mathcal{C}, \text{Cov}_1)$. If also for every $\mathcal{U} \in \text{Cov}_2$ there exists a $\mathcal{V} \in \text{Cov}_1$ such that $\mathcal{V}$ refines $\mathcal{U}$, then the categories of sheaves are equal.

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