Lemma 4.1.13. Let \(D\) be pointed and let \(C\) have a terminal object. Then the forgetful functor \(\fgt \colon C_* \to C\) induces an equivalence \[ \fgt _* \colon \Fun _*(D,C_*) \iso \Fun _*(D,C). \] In particular, the functor \((-)_* \colon \Cat _{\infty }^{\term } \to \Cat _{\infty }^{\pt }\) is right adjoint to the inclusion \(\Cat _{\infty }^{\pt } \hookrightarrow \Cat _{\infty }^{\term }\).
Proof. Since \(C_* = C_{*/}\), there is an equivalence \[ \Fun _*(D,C_*) \simeq \Fun _*(D,C)_*. \] Indeed, a natural transformation \(\const _* \to F\) is necessarily an isomorphism at the zero object as soon as \(F\) preserves terminal objects. By Corollary 4.1.12, Lemma 4.1.9, the forgetful functor from the right-hand side to \(\Fun _*(D,C)\) is an equivalence. Passing to underlying animae exhibits \((-)_*\) as a right adjoint to the inclusion. □
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