Remark 9.4.1.10. Let $\mathbb {K}$ be a collection of simplicial sets and let $H: \operatorname{\mathcal{C}}\rightarrow \widehat{\operatorname{\mathcal{C}}}$ be a functor of $\infty $-categories which exhibits $\widehat{\operatorname{\mathcal{C}}}$ as a $\mathbb {K}$-cocompletion of $\operatorname{\mathcal{C}}$. Then the diagram
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{C}}\times \Delta ^1 \ar [rr]^-{ H \times \operatorname{id}} & & \widehat{\operatorname{\mathcal{C}}} \times \Delta ^1 \ar [dl] \\ & \Delta ^1 & } \]
exhibits $\widehat{\operatorname{\mathcal{C}}} \times \Delta ^1$ as a fiberwise cocompletion of $\operatorname{\mathcal{C}}\times \Delta ^1$. Conditions $(1)$ and $(3)$ of Definition 9.4.1.8 are immediate, and conditions $(2)$ and $(4)$ follow from Variant 8.4.6.9.