Proposition 8.4.7.1. Let $h: \operatorname{\mathcal{C}}\rightarrow \widehat{\operatorname{\mathcal{C}}}$ be a functor of $\infty $-categories which exhibits $\widehat{\operatorname{\mathcal{C}}}$ as a cocompletion of $\operatorname{\mathcal{C}}$, let $\mathscr {F} \in \widehat{\operatorname{\mathcal{C}}}$ be an object, and let
\[ \xymatrix@R =50pt@C=50pt{ \widetilde{\operatorname{\mathcal{C}}} \ar [r]^-{\widetilde{h}} \ar [d] & \widehat{\operatorname{\mathcal{C}}}_{ / \mathscr {F} } \ar [d] \\ \operatorname{\mathcal{C}}\ar [r]^-{h} & \widehat{\operatorname{\mathcal{C}}} } \]
be a categorical pullback square of $\infty $-categories. Then $\widetilde{h}$ exhibits $\widehat{\operatorname{\mathcal{C}}}_{ / \mathscr {F} }$ as a cocompletion of $\widetilde{\operatorname{\mathcal{C}}}$.