Lemma 9.4.6.12. Let $\kappa $ be an uncountable regular cardinal, let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be an inner fibration of simplicial sets which is essentially $\kappa $-small, 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 exhibits $\widehat{\operatorname{\mathcal{E}}}$ as a fiberwise $\kappa $-cocompletion of $\operatorname{\mathcal{E}}$. Let $\overline{F}: \operatorname{\mathcal{C}}' \rightarrow \operatorname{\mathcal{C}}$ be a Morita equivalence of simplicial sets. If $\widehat{U}$ is a cartesian fibration, then the projection map $\operatorname{\mathcal{C}}' \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{E}}$ is also a Morita equivalence of simplicial sets.
Proof.
Set $\operatorname{\mathcal{E}}' = \operatorname{\mathcal{C}}' \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}$ and $\widehat{\operatorname{\mathcal{E}}}' = \operatorname{\mathcal{C}}' \times _{\operatorname{\mathcal{C}}} \widehat{\operatorname{\mathcal{E}}}$, so that $\overline{F}$ induces projection maps $F: \operatorname{\mathcal{E}}' \rightarrow \operatorname{\mathcal{E}}$ and $\widehat{F}: \widehat{\operatorname{\mathcal{E}}}' \rightarrow \widehat{\operatorname{\mathcal{E}}}$. We wish to show that, for every idempotent complete $\infty $-category $\operatorname{\mathcal{D}}$, precomposition with $F$ induces an equivalence of $\infty $-categories $\operatorname{Fun}( \operatorname{\mathcal{E}}, \operatorname{\mathcal{D}}) \rightarrow \operatorname{Fun}( \operatorname{\mathcal{E}}', \operatorname{\mathcal{D}})$. By virtue of Proposition 9.4.5.10, we may assume without loss of generality that $\operatorname{\mathcal{D}}$ is $\kappa $-cocomplete (in fact, we can assume that $\operatorname{\mathcal{D}}= \operatorname{\mathcal{S}}^{< \lambda }$ for some uncountable regular cardinal $\lambda \geq \kappa $). Let $\operatorname{Fun}^{\kappa }( \widehat{\operatorname{\mathcal{E}}}, \operatorname{\mathcal{D}})$ denote the full subcategory of $\operatorname{Fun}( \widehat{\operatorname{\mathcal{E}}}, \operatorname{\mathcal{D}})$ spanned by those diagrams $G: \widehat{\operatorname{\mathcal{E}}} \rightarrow \operatorname{\mathcal{D}}$ having the property that, for each vertex $C \in \operatorname{\mathcal{C}}$, the functor $G_{C} = G|_{ \widehat{\operatorname{\mathcal{E}}}_{C} }$ preserves $\kappa $-small colimits, and define $\operatorname{Fun}^{\kappa }( \widehat{\operatorname{\mathcal{E}}}', \operatorname{\mathcal{D}}) \subseteq \operatorname{Fun}( \widehat{\operatorname{\mathcal{E}}}, \operatorname{\mathcal{D}})$ similarly. We then have a commutative diagram of $\infty $-categories
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{Fun}^{\kappa }( \widehat{\operatorname{\mathcal{E}}}, \operatorname{\mathcal{D}}) \ar [r]^-{ \circ \widehat{F} } \ar [d] & \operatorname{Fun}^{\kappa }( \widehat{\operatorname{\mathcal{E}}}', \operatorname{\mathcal{D}}) \ar [d] \\ \operatorname{Fun}( \operatorname{\mathcal{E}}, \operatorname{\mathcal{D}}) \ar [r]^-{\circ F} & \operatorname{Fun}( \operatorname{\mathcal{E}}', \operatorname{\mathcal{D}}), } \]
where the vertical maps are equivalences by virtue of the universal property of fiberwise $\kappa $-cocompletions (Theorem 9.4.1.20). It will therefore suffice to show that precomposition with $\widehat{F}$ induces an equivalence of $\infty $-categories $\operatorname{Fun}^{\kappa }( \widehat{\operatorname{\mathcal{E}}}, \operatorname{\mathcal{D}}) \rightarrow \operatorname{Fun}^{\kappa }( \widehat{\operatorname{\mathcal{E}}}', \operatorname{\mathcal{D}})$. Since $\overline{F}$ is a Morita equivalence and $\widehat{U}$ is a cartesian fibration, the morphism $\widehat{F}$ is also a Morita equivalence (Proposition 9.4.5.11). In particular, precomposition with $\widehat{F}$ induces an equivalence of $\infty $-categories $\operatorname{Fun}( \widehat{\operatorname{\mathcal{E}}}, \operatorname{\mathcal{D}}) \rightarrow \operatorname{Fun}( \widehat{\operatorname{\mathcal{E}}}', \operatorname{\mathcal{D}})$. To complete the proof, it will suffice to show that an object $G \in \operatorname{Fun}( \widehat{\operatorname{\mathcal{E}}}, \operatorname{\mathcal{D}})$ belongs to the subcategory $\operatorname{Fun}^{\kappa }( \widehat{\operatorname{\mathcal{E}}}, \operatorname{\mathcal{D}})$ if and only if $G \circ \widehat{F}$ belongs to $\operatorname{Fun}^{\kappa }( \widehat{\operatorname{\mathcal{E}}}', \operatorname{\mathcal{D}})$. This follows from Lemma 9.4.6.11.
$\square$