Lemma 14.3.5 (Relative adjunctions on functor categories). Let \(L\colon C\rightleftarrows D\noloc R\) be an adjunction relative to \(B\). For every functor \(E \to B\), postcomposition induces an adjunction \[ L_*\colon \Fun _{/B}(E,C) \rightleftarrows \Fun _{/B}(E,D)\noloc R_*. \]
Proof. The counit in Definition 14.3.3 and the corresponding unit of Lemma 14.3.4 may both be whiskered with a functor \(E \to C\) or \(E \to D\) over \(B\). They therefore give a unit and counit between the displayed functors, with triangle identifications inherited from those of \(L\dashv R\). The result follows from Proposition 21.1.2. □
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