Lemma 9.4.2.6. Let $\mathbb {K}$ be a collection of simplicial sets and let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a cocartesian fibration of simplicial sets. Then there exists a commutative diagram
9.38
\begin{equation} \begin{gathered}\label{equation:describe-relative-cocompletion2} \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{E}}\ar [rr]^-{H} \ar [dr]_{U} & & \widehat{\operatorname{\mathcal{E}}} \ar [dl]^{\widehat{U}} \\ & \operatorname{\mathcal{C}}. & } \end{gathered} \end{equation}
which satisfies the conditions of Theorem 9.4.2.1.
Proof.
Choose an uncountable regular cardinal $\kappa $ such that $U$ is essentially $\kappa $-small and each $K \in \mathbb {K}$ is $\kappa $-small. Using Theorem 8.6.5.1 (and Proposition 8.6.4.15), we can choose a cocartesian fibration $U^{\vee }: \operatorname{\mathcal{E}}^{\vee } \rightarrow \operatorname{\mathcal{C}}$ and a diagram $\mathscr {K}: \operatorname{\mathcal{E}}^{\vee } \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{S}}^{<\kappa }$ which exhibits $U^{\vee }$ as a cocartesian dual of $U$. Let $\mathbb {K}'$ be the collection of all $\kappa $-small simplicial sets and set $\widehat{\operatorname{\mathcal{E}}'} = \operatorname{Fun}( \operatorname{\mathcal{E}}^{\vee } / \operatorname{\mathcal{C}}, \operatorname{\mathcal{S}}^{< \kappa } )$, so that Lemma 9.4.2.4 supplies a diagram
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{E}}\ar [rr]^-{H'} \ar [dr]_{U} & & \widehat{\operatorname{\mathcal{E}}}' \ar [dl]^{ \widehat{U}' } \\ & \operatorname{\mathcal{C}}& } \]
which satisfies the conditions of Theorem 9.4.2.1 with respect to $\mathbb {K}'$.
For each object $C \in \operatorname{\mathcal{C}}$, let $\widehat{\operatorname{\mathcal{E}}}_{C}$ denote smallest full subcategory of $\widehat{\operatorname{\mathcal{E}}}$ which contains the essential image of the functor $H'_{C}: \operatorname{\mathcal{E}}_{C} \rightarrow \widehat{\operatorname{\mathcal{E}}}'_{C}$ and is closed under $K$-indexed colimits for each $K \in \mathbb {K}$. For each edge $f: C \rightarrow D$ of the simplicial set $\operatorname{\mathcal{C}}$, Remark 5.2.2.14 guarantees that the diagram
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{E}}_{C} \ar [r] \ar [d]^{ H'_ C } & \operatorname{\mathcal{E}}_{D} \ar [d]^{ H'_ D } \\ \widehat{\operatorname{\mathcal{E}}}'_{C} \ar [r]^-{f_{!}} & \widehat{\operatorname{\mathcal{E}}}'_{D} } \]
commutes up to isomorphism, where the horizontal maps are given by covariant transport along $f$. In particular, the covariant transport functor $f_{!}: \widehat{\operatorname{\mathcal{E}}}'_{C} \rightarrow \widehat{\operatorname{\mathcal{E}}}'_{D}$ carries the essential image of $H'_{C}$ into the essential image of $H'_{D}$. Since $f_{!}$ preserves $\kappa $-small colimits, it carries $\widehat{\operatorname{\mathcal{E}}}_{C}$ into $\widehat{\operatorname{\mathcal{E}}}_{D}$.
Let $\widehat{\operatorname{\mathcal{E}}}$ denote the full simplicial subset of $\widehat{\operatorname{\mathcal{E}}}'$ spanned by those vertices $X$ which belong to $\widehat{\operatorname{\mathcal{E}}}_{C}$, where $C = \widehat{U}'(X)$. Applying Proposition 5.1.4.17, we see that $\widehat{U} = \widehat{U}'|_{ \widehat{\operatorname{\mathcal{E}}} }$ is a cocartesian fibration from $\widehat{\operatorname{\mathcal{E}}}$ to $\operatorname{\mathcal{C}}$, and that an edge of $\widehat{\operatorname{\mathcal{E}}}$ is $\widehat{U}$-cocartesian if and only if it is $\widehat{U}'$-cocartesian when regarded as an edge of $\widehat{\operatorname{\mathcal{E}}}'$. It follows that the commutative diagram
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{E}}\ar [rr]^-{H} \ar [dr]_{U} & & \widehat{\operatorname{\mathcal{E}}} \ar [dl]^{\widehat{U}} \\ & \operatorname{\mathcal{C}}& } \]
automatically satisfies conditions $(b)$ and $(d)$ of Theorem 9.4.2.1. Moreover, for each edge $f: C \rightarrow D$ of $\operatorname{\mathcal{C}}$, the associated covariant transport functor $\widehat{\operatorname{\mathcal{E}}}_{C} \rightarrow \widehat{\operatorname{\mathcal{E}}}_{D}$ is can be identified with the restriction of $f_{!}$ to $\widehat{\operatorname{\mathcal{E}}}_{C}$, and therefore preserves $K$-indexed colimits for each $K \in \mathbb {K}$. To complete the proof, it will suffice to show that for every vertex $C \in \operatorname{\mathcal{C}}$, the induced map $H_{C}: \operatorname{\mathcal{E}}_{C} \rightarrow \widehat{\operatorname{\mathcal{E}}}_{C}$ exhibits $\widehat{\operatorname{\mathcal{E}}}_{C}$ as a $\mathbb {K}$-cocompletion of $\operatorname{\mathcal{E}}_{C}$. This follows from Proposition 8.4.5.7.
$\square$