Lemma 7.38.2. Let $\mathcal{C}$ be a site and let $\{ p_ i\} _{i\in I}$ be a conservative family of points. Then

Given any map of sheaves $\varphi : \mathcal{F} \to \mathcal{G}$ we have $\forall i, \varphi _{p_ i}$ injective implies $\varphi $ injective.

Given any map of sheaves $\varphi : \mathcal{F} \to \mathcal{G}$ we have $\forall i, \varphi _{p_ i}$ surjective implies $\varphi $ surjective.

Given any pair of maps of sheaves $\varphi _1, \varphi _2 : \mathcal{F} \to \mathcal{G}$ we have $\forall i, \varphi _{1, p_ i} = \varphi _{2, p_ i}$ implies $\varphi _1 = \varphi _2$.

Given a finite diagram $\mathcal{G} : \mathcal{J} \to \mathop{\mathit{Sh}}\nolimits (\mathcal{C})$, a sheaf $\mathcal{F}$ and morphisms $q_ j : \mathcal{F} \to \mathcal{G}_ j$ then $(\mathcal{F}, q_ j)$ is a limit of the diagram if and only if for each $i$ the stalk $(\mathcal{F}_{p_ i}, (q_ j)_{p_ i})$ is one.

Given a finite diagram $\mathcal{F} : \mathcal{J} \to \mathop{\mathit{Sh}}\nolimits (\mathcal{C})$, a sheaf $\mathcal{G}$ and morphisms $e_ j : \mathcal{F}_ j \to \mathcal{G}$ then $(\mathcal{G}, e_ j)$ is a colimit of the diagram if and only if for each $i$ the stalk $(\mathcal{G}_{p_ i}, (e_ j)_{p_ i})$ is one.

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