Proposition 7.1.5.10. Let $U: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories and let $f: X \rightarrow Y$ be a morphism in $\operatorname{\mathcal{C}}$ with the property that $U(f)$ is an isomorphism. Then any two of the following three conditions imply the third:
The object $X$ is $U$-initial.
The object $Y$ is $U$-initial.
The morphism $f$ is an isomorphism.
Proof. Fix an object $Z \in \operatorname{\mathcal{C}}$. We claim that any two of the following three conditions imply the third:
The functor $U$ induces a homotopy equivalence $\operatorname{Hom}_{\operatorname{\mathcal{C}}}( X,Z) \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{C}}}( U(X), U(Z) )$.
The functor $U$ induces a homotopy equivalence $\operatorname{Hom}_{\operatorname{\mathcal{C}}}( Y,Z) \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{C}}}( U(Y), U(Z) )$.
Precomposition $[f]$ induces a homotopy equivalence $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(Y,Z) \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Z)$ (see Notation 4.6.9.15).
This follows from the commutativity of the diagram
in the homotopy category $\mathrm{h} \mathit{\operatorname{Kan}}$, since the bottom horizontal map is a homotopy equivalence (by virtue of our assumption that $U(f)$ is an isomorphism). Proposition 7.1.5.10 follows by allowing the object $Z$ to vary. $\square$