Remark 9.4.1.15. Let $\mathbb {K}$ be a collection of simplicial sets and suppose we are given a commutative diagram of simplicial sets
which exhibits $\widehat{\operatorname{\mathcal{E}}}$ as a fiberwise $\mathbb {K}$-cocompletion of $\operatorname{\mathcal{E}}$. Let $\operatorname{\mathcal{E}}' \subseteq \widehat{\operatorname{\mathcal{E}}}$ be the full simplicial subset spanned by those vertices which belong to the image of $H$. It follows from Remark 9.4.1.14 and Proposition 5.1.7.9 that the induced map $\operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{E}}'$ is an equivalence of inner fibrations over $\operatorname{\mathcal{C}}$, so that the inclusion map $\operatorname{\mathcal{E}}' \hookrightarrow \widehat{\operatorname{\mathcal{E}}}$ exhibits $\widehat{\operatorname{\mathcal{E}}}$ as a fiberwise $\mathbb {K}$-cocompletion of $\operatorname{\mathcal{E}}'$.