The Stacks project

Lemma 6.27.3. Let $X$ be a topological space, and let $x \in X$ a point. The functors $\mathcal{F} \mapsto \mathcal{F}_ x$ and $A \mapsto i_{x, *}A$ are adjoint. In a formula

\[ \mathop{\mathrm{Mor}}\nolimits _{\textit{Sets}}(\mathcal{F}_ x, A) = \mathop{\mathrm{Mor}}\nolimits _{\mathop{\mathit{Sh}}\nolimits (X)}(\mathcal{F}, i_{x, *}A). \]

A similar statement holds for the case of abelian groups, algebraic structures. In the case of sheaves of modules we have

\[ \mathop{\mathrm{Hom}}\nolimits _{\mathcal{O}_{X, x}}(\mathcal{F}_ x, A) = \mathop{\mathrm{Hom}}\nolimits _{\mathcal{O}_ X}(\mathcal{F}, i_{x, *}A). \]

Proof. Omitted. Hint: The stalk functor can be seen as the pullback functor for the morphism $i_ x : \{ x\} \to X$. Then the adjointness follows from adjointness of $i_ x^{-1}$ and $i_{x, *}$ (resp. $i_ x^*$ and $i_{x, *}$ in the case of sheaves of modules). $\square$

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