Corollary 4.1.12. Let \(D\) be a pointed \(\infty \)-category and let \(C\) be an \(\infty \)-category with a terminal object. Then the \(\infty \)-category \(\Fun _*(D,C)\) is pointed.
Proof. The constant functor \(\const _*\colon D \to C\) is terminal in \(\Fun (D,C)\), hence also in the full subcategory \(\Fun _*(D,C)\). We claim that it is also initial. Let \(0 \in D\) denote a zero object, and let \(F\colon D \to C\) be a functor preserving terminal objects. Since \(0\) is also initial in \(D\), limits over \(D\) are given by evaluation at \(0\) by Lemma 21.2.4. We therefore get \[ \Hom _{\Fun _*(D,C)}(\const _*,F) \simeq \Hom _C(*,\lim _{d \in D} F(d)) \simeq \Hom _C(*,F(0)). \] Since \(F\) preserves terminal objects and \(0\) is terminal in \(D\), the object \(F(0)\) is terminal in \(C\). The last hom anima is therefore contractible, proving that \(\const _*\) is initial. □
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