Kerodon

Proof. Let $\operatorname{\mathcal{E}}$ denote the oriented fiber product $\operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{ \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}_{< \lambda } ) } \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}_{< \lambda } )$, and let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ and $V: \operatorname{\mathcal{E}}\rightarrow \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}_{< \lambda } )$ be the projection maps. For each object $\mathscr {F} \in \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}_{< \lambda } )$, let $\operatorname{\mathcal{E}}_{\mathscr {F}}$ denote the fiber $V^{-1} \{ \mathscr {F} \} $, and let $U_{\mathscr {F}}: \operatorname{\mathcal{E}}_{\mathscr {F}} \rightarrow \operatorname{\mathcal{C}}$ be the restriction of $U$. Then $U_{\mathscr {F}}$ is a right fibration with contravariant transport representation $\mathscr {F}$ (Corollary 8.4.2.7). In particular, the right fibration $U_{\mathscr {F}}$ is essentially $\lambda $-small, so the $\infty $-category $\operatorname{\mathcal{E}}_{\mathscr {F}}$ is essentially $\lambda $-small (Proposition 4.7.9.10). It follows that $V$ is a cocartesian fibration (Corollary 5.3.7.3) which admits a covariant transport representation

We will complete the proof by showing that the functor $T$ is $\lambda $-cocontinuous.

Fix a $\lambda $-small $\infty $-category $\operatorname{\mathcal{K}}$ and a (levelwise) colimit diagram $\overline{G}: \operatorname{\mathcal{K}}^{\triangleright } \rightarrow \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}_{< \lambda } )$; we wish to show that $T \circ \overline{G}$ is a colimit diagram in $\operatorname{\mathcal{QC}}_{< \lambda }$. Set $\operatorname{\mathcal{E}}' = \operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{ \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}_{< \lambda } ) } \operatorname{\mathcal{K}}^{\triangleright }$, so that $T \circ \overline{G}$ is a covariant transport representation for the projection map $V': \operatorname{\mathcal{E}}' \rightarrow \operatorname{\mathcal{K}}^{\triangleright }$. Let $z$ denote the cone point of $\operatorname{\mathcal{K}}^{\triangleright } = \operatorname{\mathcal{K}}\star \{ z\} $ and define

so that the cocartesian fibration $V'$ admits a covariant refraction diagram $\mathrm{Rf}: \operatorname{\mathcal{E}}'_0 \rightarrow \operatorname{\mathcal{E}}'_{z}$ (see Definition 7.4.5.8). Let $U': \operatorname{\mathcal{E}}' \rightarrow \operatorname{\mathcal{C}}$ be projection onto the first factor and let $W$ be the collection of all morphisms $e$ of $\operatorname{\mathcal{E}}'_0$ such that $U'(e)$ is an identity morphism in $\operatorname{\mathcal{C}}$. It follows from Corollary 5.3.7.3 that a morphism $e$ of $\operatorname{\mathcal{E}}'_0$ is $V'$-cocartesian if and only if $U(e)$ is an isomorphism in $\operatorname{\mathcal{C}}$: that is, if and only if $e$ is isomorphic (as an object of $\operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{E}}'_0 )$) to an element of $W$. By virtue of Theorem 7.4.5.13, it will suffice to show that the functor $\mathrm{Rf}$ exhibits the $\infty $-category $\operatorname{\mathcal{E}}'_{v}$ as a localization of $\operatorname{\mathcal{E}}'_0$ with respect to $W$.

Let $h: \Delta ^1 \times \operatorname{\mathcal{E}}'_0 \rightarrow \operatorname{\mathcal{E}}'$ be a natural transformation which exhibits $\mathrm{Rf}$ as a covariant refraction diagram for $V'$ (see Definition 7.4.5.8). Then $h$ carries each object $E \in \operatorname{\mathcal{E}}'_0$ to a $V'$-cocartesian morphism $h_{E}: E \rightarrow \mathrm{Rf}(E)$, so that $U'( h_{E} )$ is an isomorphism in $\operatorname{\mathcal{C}}$. We may therefore assume (replacing $\mathrm{Rf}$ by an isomorphic functor if necessary) that the composition $(U' \circ h): \Delta ^1 \times \operatorname{\mathcal{E}}'_0 \rightarrow \operatorname{\mathcal{C}}$ factors through $\operatorname{\mathcal{E}}'_0$ (so that each $U'(h_{E})$ is an identity morphism in $\operatorname{\mathcal{C}}$). Then, for each $C \in \operatorname{\mathcal{C}}$, $\mathrm{Rf}$ restricts to a functor

which is a covariant refraction diagram for the left fibration $\{ C\} \operatorname{\vec{\times }}_{\operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}_{< \lambda } )} \operatorname{\mathcal{K}}^{\triangleright } \rightarrow \operatorname{\mathcal{K}}^{\triangleright }$. Our assumption that $\overline{G}$ is a (levelwise) colimit diagram then guarantees that $\mathrm{Rf}_{C}$ is a weak homotopy equivalence: that is, it exhibits the Kan complex $\{ C\} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}'_{z}$ as a localization of $\{ C\} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}'_0$ with respect to the collection $W_{C}$ of all morphisms of $\{ C\} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}'_0$ (Theorem 7.4.5.13). Applying Theorem 6.3.4.1, we conclude that $\mathrm{Rf}$ exhibits $\operatorname{\mathcal{E}}'_{v}$ as a localization of $\operatorname{\mathcal{E}}'_0$ with respect to the collection of morphisms $W = \bigcup _{C \in \operatorname{\mathcal{C}}} W_{C}$, as desired. $\square$