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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