Corollary 4.5.3.10 (023C)

Corollary 4.5.3.10. Let $\{ f_ i: X_{i} \rightarrow Y_ i \} _{i \in I}$ be a collection of categorical equivalences indexed by a set $I$. Then the coproduct map

\[ f: \coprod _{i \in I} X_{i} \rightarrow \coprod _{i \in I} Y_ i \]

is also a categorical equivalence.

Proof. By virtue of Proposition 4.5.3.8, it will suffice to show that for every $\infty $-category $\operatorname{\mathcal{C}}$, precomposition with $f$ induces an equivalence of $\infty $-categories

\[ F: \operatorname{Fun}( \coprod _{i \in I} Y_ i, \operatorname{\mathcal{C}}) \rightarrow \operatorname{Fun}( \coprod _{i \in I} X_ i, \operatorname{\mathcal{C}}). \]

Note that $F$ factors as a product of functors $F_ i: \operatorname{Fun}(Y_ i, \operatorname{\mathcal{C}}) \rightarrow \operatorname{Fun}(X_ i, \operatorname{\mathcal{C}})$, each of which is induced by precomposition with $f_{i}$. Since each $f_{i}$ is a categorical equivalence, Proposition 4.5.3.8 guarantees that each $F_{i}$ is an equivalence of $\infty $-categories. Applying Remark 4.5.1.17, we conclude that $F$ is an equivalence of $\infty $-categories. $\square$