Remark 7.7.4.4. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category which contains an initial object $\emptyset $, and let $X_0$ and $X_1$ be objects of $\operatorname{\mathcal{C}}$. Then an object $X$ of $\operatorname{\mathcal{C}}$ is a coproduct of $X_0$ with $X_1$ if and only if there exists a pushout square
7.86
\begin{equation} \begin{gathered}\label{equation:pushout-also-pullback} \xymatrix { \emptyset \ar [r] \ar [d] & X_0 \ar [d]^{f_0} \\ X_1 \ar [r]^{f_1} & X } \end{gathered} \end{equation}
(see Corollary 7.6.2.19). In this case, the coproduct is disjoint if and only if (7.86) is also a pullback square, and the morphisms $f_ i$ and $f_ j$ are monomorphisms.