Lemma 4.24.2. Let $u : \mathcal{C} \to \mathcal{D}$ be a functor between categories. If for each $y \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{D})$ the functor $x \mapsto \mathop{\mathrm{Mor}}\nolimits _\mathcal {D}(u(x), y)$ is representable, then $u$ has a right adjoint.
Proof. For each $y$ choose an object $v(y)$ and an isomorphism $\mathop{\mathrm{Mor}}\nolimits _\mathcal {C}(-, v(y)) \to \mathop{\mathrm{Mor}}\nolimits _\mathcal {D}(u(-), y)$ of functors. By Yoneda's lemma (Lemma 4.3.5) for any morphism $g : y \to y'$ the transformation of functors
\[ \mathop{\mathrm{Mor}}\nolimits _\mathcal {C}(-, v(y)) \to \mathop{\mathrm{Mor}}\nolimits _\mathcal {D}(u(-), y) \to \mathop{\mathrm{Mor}}\nolimits _\mathcal {D}(u(-), y') \to \mathop{\mathrm{Mor}}\nolimits _\mathcal {C}(-, v(y')) \]
corresponds to a unique morphism $v(g) : v(y) \to v(y')$. We omit the verification that $v$ is a functor and that it is right adjoint to $u$. $\square$
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