Notation 7.6.1.13 (Diagonal Morphisms). Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, let $X$ be an object of $\operatorname{\mathcal{C}}$, and suppose that there exists a diagram

\[ X \xleftarrow { \pi } X \times X \xrightarrow {\pi '} X \]

which exhibits $X \times X$ as a product of $X$ with itself. Then there exists a morphism $\delta : X \rightarrow X \times X$ with the property that the compositions $\pi \circ \delta $ and $\pi ' \circ \delta $ are homotopic to $\operatorname{id}_{X}$. Moreover, the morphism $\delta $ is uniquely determined up to homotopy. For this reason, we will often denote $\delta $ by $\delta _{X}$ and refer to it as the *diagonal morphism* of the object $X$.