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 $\mathcal{C}$.

1. 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.

2. 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.

Proof. Part (1) follows directly from Lemma 7.8.4 and the definitions.

Proof of (2). Let $\mathcal{F}$ be a sheaf of sets for the site $(\mathcal{C}, \text{Cov}_2)$. Let $\mathcal{U} \in \text{Cov}_1$, say $\mathcal{U} = \{ U_ i \to U\} _{i \in I}$. By assumption we may choose a refinement $\mathcal{V} \in \text{Cov}_2$ of $\mathcal{U}$, say $\mathcal{V} = \{ V_ j \to U\} _{j \in J}$ and refinement given by $\alpha : J \to I$ and $f_ j : V_ j \to U_{\alpha (j)}$. Observe that $\mathcal{F}$ satisfies the sheaf condition for $\mathcal{V}$ and for the coverings $\{ V_ j \times _ U U_ i \to U_ i\} _{j \in J}$ as these are in $\text{Cov}_2$. Hence $\mathcal{F}$ satisfies the sheaf condition for $\mathcal{U}$ by Lemma 7.8.6. $\square$

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