Lemma 21.5.5. Consider an object \(j\) of an \(\infty \)-category \(J\). Then \(j\colon * \to C\) is an initial functor if and only if \(j\) is an initial object, and it is a final functor if and only if \(j\) is a final (i.e. terminal) object of \(J\).
Proof. Given an object \(x\) of \(C\), unwinding definitions reveals that the relative slice category of the functor \(j\colon * \to C\) is equivalent to the hom anima \(\Hom _C(j,x)\). Since this is already an anima, it is equivalent to its geometric realization, and hence it is weakly contractible if and only if it is contractible. So we see that \(j\) is an initial functor if and only if the hom anima \(\Hom _C(j,x)\) is contractible for every object \(x\) of \(C\), which by is the case by definition precisely if \(j\) is an initial object. □
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