Lemma 21.1.7. Let \(F\colon C \rightleftarrows D \noloc G\) be an adjunction of \(\infty \)-categories. Then the induced maps \(\geom {F}\colon \geom {C} \to \geom {D}\) and \(\geom {G}\colon \geom {D} \to \geom {C}\) are inverse equivalences.
Proof. Thinking of the counit \(\epsilon \colon FG \to \id _D\) as a functor \([1] \times D \to D\), it induces another functor \[ [1] \times \geom {D} \to \geom {[1]} \times \geom {D} \overset {\text{1.5.24}}{\simeq } \geom {[1] \times D} \xrightarrow {\geom {\epsilon }} \geom {D}, \] i.e. a natural transformation \(\geom {F} \geom {G} = \geom {FG} \to \geom {\id _D} = \id _{\geom {D}}\). Since \(\geom {D}\) is an anima, this natural transformation is automatically a natural isomorphism by Axiom J. Similarly, the unit \(\eta \colon \id _C \to GF\) induces a natural transformation \(\geom {\eta }\colon \id _{\geom {C}} \to \geom {G} \geom {F}\). This shows that \(\geom {G}\) is inverse to \(\geom {F}\), as desired. □
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