Proposition 23.6.7. Let \(F\colon I \to C\) be a functor of \(\infty \)-categories. A natural transformation \(\eta \colon F \to \const _x\) exhibits \(x \in C\) as the colimit of \(F\) if and only if the natural map \[ \eta ^*\colon \Hom _C(x,y) \iso \lim _{i \in I\catop } \Hom _C(F(i),y) \] obtained by applying \(\Hom _C(-,y)\) to \(\eta \) is an equivalence for all \(y \in C\). A similar assertion holds for limits.
Proof. Consider the following commutative diagram:
here the right vertical map is induced by the evaluation functors \(\ev _i\colon \Fun (I,C) \to C\). By definition, \(x\) is a colimit of \(F\) if and only if the top map is an equivalence for all \(y\), hence it will suffice to show that the right vertical map is an equivalence. By Proposition 23.6.6, the top right is equivalent to \(\lim _{(i \to i') \in \Tw (I)} \Hom _C(F(i), y)\). It thus remains to show that the forgetful functor \(s\colon \Tw (I) \to I\catop \) is final. By Theorem 21.5.1, it suffices to show that the relative slices are weakly contractible. But these relative slices are weakly equivalent [explain]1 to the actual slices \(I_{i/}\), which have initial objects, so the claim follows from Lemma 21.5.6. □
Notes
1There is an adjunction \(\Tw (I)_{i/} \rightleftarrows I_{i/}\), and we have Lemma 21.1.7
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