Lemma 6.21.4. Let $f : X \to Y$ be a continuous map. Let $x \in X$. Let $\mathcal{G}$ be a presheaf of sets on $Y$. There is a canonical bijection of stalks $(f_ p\mathcal{G})_ x = \mathcal{G}_{f(x)}$.

Proof. This you can see as follows

\begin{eqnarray*} (f_ p\mathcal{G})_ x & = & \mathop{\mathrm{colim}}\nolimits _{x \in U} f_ p\mathcal{G}(U) \\ & = & \mathop{\mathrm{colim}}\nolimits _{x \in U} \mathop{\mathrm{colim}}\nolimits _{f(U) \subset V} \mathcal{G}(V) \\ & = & \mathop{\mathrm{colim}}\nolimits _{f(x) \in V} \mathcal{G}(V) \\ & = & \mathcal{G}_{f(x)} \end{eqnarray*}

Here we have used Categories, Lemma 4.14.9, and the fact that any $V$ open in $Y$ containing $f(x)$ occurs in the third description above. Details omitted. $\square$

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