Lemma 21.1.6. Let \(F \dashv G\) be an adjunction of \(\infty \)-categories. Then:
- (1)
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For every \(\infty \)-category \(E\) the functors \[ G^*\colon \Fun (C,E) \rightleftarrows \Fun (D,E) \noloc F^* \] form an adjunction, with unit and counit given by \[ G^*F^* = (FG)^* \xrightarrow {\epsilon ^*} \id _D^* = \id _{\Fun (D,E)} \qquadtext { and } \id _{\Fun (C,E)} = \id _C^* \xrightarrow {\eta ^*} (GF)^* = F^*G^*. \]
- (2)
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For every \(\infty \)-category \(E\) the functors \[ F_*\colon \Fun (E,C) \rightleftarrows \Fun (E,D) \noloc G_* \] form an adjunction, with unit and counit given by \[ F_*G_* = (FG)_* \xrightarrow {\epsilon _*} (\id _D)_* = \id _{\Fun (D,E)} \qquadtext { and } \id _{\Fun (C,E)} = (\id _C)_* \xrightarrow {\eta _*} (GF)_* = G_*F_*. \]
Proof. The triangle identities for \(\epsilon ^*\) and \(\eta ^*\), resp. \(\epsilon _*\) and \(\eta _*\), follow immediately from the triangle identities for \(\epsilon \) and \(\eta \). We leave the details to the reader. □
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