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Proposition 5.4.6.10 (Uniqueness). Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be an equivalence of $\infty$-categories. If $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ are minimal, then $F$ is an isomorphism of simplicial sets.

Proof. Let $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$ be a homotopy inverse to $F$. It follows from Lemma 5.4.6.8 that $F$ and $G$ are monomorphisms of simplicial sets. We will complete the proof by showing that the composite map $(F \circ G): \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{D}}$ is an epimorphism of simplicial sets (so that, in particular, $F$ is an epimorphism). Let $\sigma$ be an $n$-simplex of $\operatorname{\mathcal{D}}$; we wish to show that $\sigma$ belongs to the image of $F \circ G$. The proof proceeds by induction on $n$. Set $\sigma _0 = \sigma |_{ \operatorname{\partial \Delta }^ n }$; our inductive hypothesis then guarantees that we can write $\sigma _0 = (F \circ G)( \tau _0 )$ for some morphism $\tau _0: \operatorname{\partial \Delta }^ n \rightarrow \operatorname{\mathcal{D}}$.

Choose a $2$-simplex

$\xymatrix@R =50pt@C=50pt{ F \circ G \ar [dr]_{\alpha } \ar [rr]^-{ \operatorname{id}_{ F \circ G} } & & F \circ G \\ & \operatorname{id}_{\operatorname{\mathcal{D}}} \ar [ur]_{\beta } & }$

in the $\infty$-category $\operatorname{Fun}(\operatorname{\mathcal{D}}, \operatorname{\mathcal{D}})$, where $\alpha$ and $\beta$ are isomorphisms. Precomposing with $\tau _0: \operatorname{\partial \Delta }^ n \rightarrow \operatorname{\mathcal{D}}$, we obtain a $2$-simplex

5.36
$$\begin{gathered}\label{equation:uniqueness-of-minimal} \xymatrix@R =50pt@C=50pt{ \sigma _0 \ar [dr]_{\alpha (\tau _0)} \ar [rr]^-{ \operatorname{id}} & & \sigma _0 \\ & \tau _0 \ar [ur]_{\beta (\tau _0)} & } \end{gathered}$$

in the $\infty$-category $\operatorname{Fun}( \operatorname{\partial \Delta }^ n, \operatorname{\mathcal{D}})$. Using Corollary 4.4.5.9, we can lift $\alpha (\tau _0)$ to an isomorphism $\widetilde{\alpha }: \sigma \rightarrow \tau$ in the $\infty$-category $\operatorname{Fun}( \Delta ^ n, \operatorname{\mathcal{D}})$. Since $\operatorname{\mathcal{D}}$ is an $\infty$-category, Theorem 1.4.6.1 guarantees that we can lift (5.36) to a $2$-simplex

$\xymatrix@R =50pt@C=50pt{ \sigma \ar [rr]^-{ \gamma } \ar [dr]_{\widetilde{\alpha }} & & (F \circ G)(\tau ) \\ & \tau . \ar [ur]_{\beta (\tau )} & }$

in the $\infty$-category $\operatorname{Fun}( \Delta ^ n, \operatorname{\mathcal{D}})$. By construction, $\gamma$ is an isomorphism whose image in $\operatorname{Fun}( \operatorname{\partial \Delta }^ n, \operatorname{\mathcal{D}})$ is an identity morphism. Our assumption that $\operatorname{\mathcal{D}}$ is minimal then guarantees that $\sigma = (F \circ G)(\tau )$ belongs to the image of $F \circ G$. $\square$