Kerodon

Proof. If $X$ and $Y$ are initial objects of $\operatorname{\mathcal{C}}$, then they are contained in the contractible Kan complex $\operatorname{\mathcal{C}}^{\mathrm{init}}$ of Corollary 4.6.7.14 and are therefore isomorphic when viewed as objects of $\operatorname{\mathcal{C}}$. Conversely, suppose that $X$ is initial and that there exists an isomorphism $f: X \rightarrow Y$ in $\operatorname{\mathcal{C}}$; we wish to show that $Y$ is also initial. Fix an object $Z \in \operatorname{\mathcal{C}}$; we wish to show that the mapping space $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(Y,Z)$ is contractible. Let us regard the homotopy category $\mathrm{h} \mathit{\operatorname{\mathcal{C}}}$ as enriched over the homotopy category $\mathrm{h} \mathit{\operatorname{Kan}}$ of Kan complexes (see Construction 4.6.9.13). Since $f$ an isomorphism in $\operatorname{\mathcal{C}}$, its homotopy class $[f]$ is an isomorphism in the homotopy category $\mathrm{h} \mathit{\operatorname{\mathcal{C}}}$, so composition with $[f]$ induces an isomorphism $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(Y,Z) \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Z)$ in the category $\mathrm{h} \mathit{\operatorname{Kan}}$. Since the Kan complex $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Z)$ is contractible, it follows that $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(Y,Z)$ is also contractible. $\square$