The Stacks project

Theorem 4.25.3 (Adjoint functor theorem). Let $G : \mathcal{C} \to \mathcal{D}$ be a functor of big categories. Assume $\mathcal{C}$ has limits, $G$ commutes with them, and for every object $y$ of $\mathcal{D}$ there exists a set of pairs $(x_ i, f_ i)_{i \in I}$ with $x_ i \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$, $f_ i \in \mathop{\mathrm{Mor}}\nolimits _\mathcal {D}(y, G(x_ i))$ such that for any pair $(x, f)$ with $x \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$, $f \in \mathop{\mathrm{Mor}}\nolimits _\mathcal {D}(y, G(x))$ there are an $i$ and a morphism $h : x_ i \to x$ such that $f = G(h) \circ f_ i$. Then $G$ has a left adjoint $F$.

Proof. The assumptions imply that for every object $y$ of $\mathcal{D}$ the functor $x \mapsto \mathop{\mathrm{Mor}}\nolimits _\mathcal {D}(y, G(x))$ satisfies the assumptions of Lemma 4.25.1. Thus it is representable by an object, let's call it $F(y)$. An application of Yoneda's lemma (Lemma 4.3.5) turns the rule $y \mapsto F(y)$ into a functor which by construction is an adjoint to $G$. We omit the details. $\square$

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