Definition 14.3.3 (Relative adjunction). Let \(L\) be an object of \(\Fun _{/B}(C,D)\). A right adjoint to \(L\) relative to \(B\) consists of an object \(R \in \Fun _{/B}(D,C)\) together with a morphism \[ \epsilon \colon LR \to \id _D \] in \(\Fun _{/B}(D,D)\) whose image in \(\Fun (D,D)\) exhibits the underlying functor of \(R\) as a right adjoint to the underlying functor of \(L\). We then call \(L\dashv R\) an adjunction relative to \(B\).
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