Lemma 7.32.4. For any functor $u : \mathcal{C} \to \textit{Sets}$. The functor $u^ p$ is a right adjoint to the stalk functor on presheaves.

Proof. Let $\mathcal{F}$ be a presheaf on $\mathcal{C}$. Let $E$ be a set. A morphism $\mathcal{F} \to u^ pE$ is given by a compatible system of maps $\mathcal{F}(U) \to \text{Map}(u(U), E)$, i.e., a compatible system of maps $\mathcal{F}(U) \times u(U) \to E$. And by definition of $\mathcal{F}_ p$ a map $\mathcal{F}_ p \to E$ is given by a rule associating with each triple $(U, x, \sigma )$ an element in $E$ such that equivalent triples map to the same element, see discussion surrounding Equation (7.32.1.1). This also means a compatible system of maps $\mathcal{F}(U) \times u(U) \to E$. $\square$

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