Corollary 7.7.4.12. Small coproducts are strongly universal in the $\infty $-categories $\operatorname{\mathcal{S}}$ and $\operatorname{\mathcal{QC}}$.
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Proof. We will give the proof for $\operatorname{\mathcal{QC}}$; the analogous statement for the $\infty $-category $\operatorname{\mathcal{S}}$ is similar (but easier). By virtue of Variant 7.7.4.8, the $\infty $-category $\operatorname{\mathcal{QC}}$ has disjoint coproducts. It will therefore suffice to show that small coproducts in $\operatorname{\mathcal{QC}}$ are universal. Let $\{ \operatorname{\mathcal{C}}_ i \} _{i \in I}$ be a small collection of $\infty $-categories and let $\operatorname{\mathcal{C}}= \coprod _{i \in I} \operatorname{\mathcal{C}}_ i$ denote their coproduct, formed in the category of simplicial sets. For each $i \in I$, let $f_ i: \operatorname{\mathcal{C}}_ i \hookrightarrow \operatorname{\mathcal{C}}$ be the inclusion map. The collection $\{ f_ i \} _{i \in I}$ determines a morphism $f: \{ \operatorname{\mathcal{C}}_ i \} _{i \in I} \rightarrow \underline{\operatorname{\mathcal{C}}}$ in the $\infty $-category $\operatorname{Fun}(I, \operatorname{\mathcal{QC}})$, which exhibits $\operatorname{\mathcal{C}}$ as a coproduct of $\{ \operatorname{\mathcal{C}}_ i \} _{i \in I}$ in the $\infty $-category $\operatorname{\mathcal{QC}}$ (Example 7.6.1.17).
Suppose we are given a functor of $\infty $-categories $\operatorname{\mathcal{C}}' \rightarrow \operatorname{\mathcal{C}}$. For each $i \in I$, let $\operatorname{\mathcal{C}}'_{i}$ denote the inverse image of $\operatorname{\mathcal{C}}_{i}$ in $\operatorname{\mathcal{C}}'$, so that we have a pullback diagram
\[ \xymatrix { \operatorname{\mathcal{C}}'_{i} \ar [r]^{ f'_{i} } \ar [d] & \operatorname{\mathcal{C}}' \ar [d] \\ \operatorname{\mathcal{C}}_ i \ar [r]^{f_ i} & \operatorname{\mathcal{C}}} \]
in the category of simplicial sets. Since $f_{i}$ is an isofibration, this diagram is also a categorical pullback square (Corollary 4.5.2.27), and therefore determines a pullback diagram in the $\infty $-category $\operatorname{\mathcal{QC}}$ (Example 7.6.3.4). Allowing $i$ to vary, we obtain a levelwise pullback diagram
\[ \xymatrix { \{ \operatorname{\mathcal{C}}'_{i} \} _{i \in I} \ar [r]^{ f' } \ar [d] & \underline{ \operatorname{\mathcal{C}}' } \ar [d] \\ \{ \operatorname{\mathcal{C}}_ i \} _{i \in I} \ar [r]^{f} & \underline{\operatorname{\mathcal{C}}} } \]
in the $\infty $-category $\operatorname{Fun}(I, \operatorname{\mathcal{QC}})$. To complete the proof, it will suffice to show thatt $f'$ exhibits $\operatorname{\mathcal{C}}'$ as a coproduct of $\{ \operatorname{\mathcal{C}}'_ i \} _{i \in I}$ in the $\infty $-category $\operatorname{\mathcal{QC}}$, which follows from Example 7.6.1.17. $\square$