Lemma 6.2.10. Let \(\gamma \colon C \to C[W^{-1}]\) be the localization functor inverting a collection of morphisms \(W\) in some \(\infty \)-category \(C\). Let \(X \in C\) be an object with the property that \(\Hom _C(X,-)\colon C \to \An \) inverts all morphisms in \(W\). Then for every \(Y \in C\) the map \[ \Hom _C(X,Y) \to \Hom _{C[W^{-1}]}(\gamma X, \gamma Y) \] is an isomorphism of animae.
Proof. The assumption on \(X\) guarantees that \(\Hom _C(X,-)\) descends to a functor \(C[W^{-1}] \to \An \). We need to show that this functor is corepresented by \(\gamma X \in C[W^{-1}]\). But this follows immediately from the full faithfulness of \(\gamma ^*\colon \Fun (C[W^{-1}],\An ) \hookrightarrow \Fun (C,\An )\): for any other functor \(F\colon C[W^{-1}] \to \An \) we have \[ \Nat _{C[W^{-1}] \to \An }(\Hom _C(X,-),F) \iso \Nat _{C \to \An }(\Hom _C(X,-),F\circ \gamma ) \iso F(\gamma (X)). \qedhere \] □
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