Lemma 9.4.3.2. Let $\mathbb {K}$ be a collection of simplicial sets, let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a cartesian fibration of simplicial sets, and suppose we are given a commutative diagram
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{E}}\ar [rr]^-{H} \ar [dr]_{U} & & \widehat{\operatorname{\mathcal{E}}} \ar [dl]^{ \widehat{U} } \\ & \operatorname{\mathcal{C}}& } \]
which satisfies the conditions of Theorem 9.4.3.1. Then $H$ exhibits $\widehat{\operatorname{\mathcal{E}}}$ as a fiberwise $\mathbb {K}$-cocompletion of $\operatorname{\mathcal{E}}$.
Proof.
Without loss of generality, we may assume that $\operatorname{\mathcal{C}}= \Delta ^1$. For $i \in \{ 0,1\} $, let $\operatorname{\mathcal{E}}_{i}$ denote the fiber $\{ i\} \times _{\Delta ^1} \operatorname{\mathcal{E}}$ and define $\widehat{\operatorname{\mathcal{E}}}_{i}$ similarly, so that $H$ restricts to a functor $H_{i}: \operatorname{\mathcal{E}}_{i} \rightarrow \widehat{\operatorname{\mathcal{E}}}_{i}$. By assumption, the functor $H_{i}$ exhibits $\widehat{\operatorname{\mathcal{E}}}_{i}$ as a $\mathbb {K}$-cocompletion of $\operatorname{\mathcal{E}}_{i}$, and is therefore fully faithful (Proposition 8.4.5.3). Since $H$ carries $U$-cartesian morphisms of $\operatorname{\mathcal{E}}$ to $\widehat{U}$-cartesian morphisms of $\widehat{\operatorname{\mathcal{E}}}$, it follows that $H$ is fully faithful (Proposition 5.1.6.7). It follows from Example 9.4.1.6 that the fibration $\widehat{U}$ is $\mathbb {K}$-cocomplete.
Fix an uncountable regular cardinal $\kappa $ such that $\widehat{\operatorname{\mathcal{E}}}$ is locally $\kappa $-small and every simplicial set $K \in \mathbb {K}$ is essentially $\kappa $-small. To complete the proof, it will suffice to show that for every object $X \in \operatorname{\mathcal{E}}_{0}$, the functor
\[ \mathscr {F}: \widehat{\operatorname{\mathcal{E}}}_{1} \rightarrow \operatorname{\mathcal{S}}^{< \kappa } \quad \quad Y \mapsto \operatorname{Hom}_{\widehat{\operatorname{\mathcal{E}}} }( H(X), Y ) \]
commutes with $K$-indexed colimits, for each $K \in \mathbb {K}$. Since $\widehat{U}$ is a cartesian fibration, this functor factors as a composition $\widehat{\operatorname{\mathcal{E}}}_{1} \xrightarrow { f^{\ast } } \widehat{\operatorname{\mathcal{E}}}_0 \xrightarrow { \mathscr {G}} \operatorname{\mathcal{S}}^{< \kappa }$, where $f^{\ast }$ is given by contravariant transport along the nondegenerate edge of $\operatorname{\mathcal{C}}$ and the functor $\mathscr {G}$ is corepresented by $H(X)$. We are therefore reduced to showing that the functor $\mathscr {G}$ preserves $K$-indexed colimits, which follows from the recognition principle of Variant 8.4.6.9.
$\square$