Corollary 21.1.5. A functor \(F\colon C \to D\) admits a right adjoint if and only if every object of \(D\) admits a right adjoint object under \(F\). Dually, \(F\) admits a left adjoint if and only if every object of \(D\) admits a left adjoint object under \(F\).
Proof. We again only prove the claim about right adjoints. If a right adjoint \(G\colon D \to C\) exists, then by Proposition 21.1.2 we see that the object \(Z := GY\) together with the counit map \(\epsilon _Y \colon FGY \to Y\) provides a right adjoint object to \(Y\) under \(F\). Conversely, if all the right adjoint objects exists, then we may apply Lemma 21.1.4 with \(D = D_R\) to obtain a functor \(G\colon D \to C\) together with a natural isomorphism \[ \Hom _C(F(-),-) \cong \Hom _{D}(-,G(-)), \] which means that \(G\) is a right adjoint to \(F\). □
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