Kerodon

Proof. Fix an object $\widetilde{Y} \in \operatorname{\mathcal{E}}_{ / X }$, which we identify with a pair $(Y,v)$ where $Y$ is an object of $\operatorname{\mathcal{E}}$ and $v: H(Y) \rightarrow X$ is a morphism in $\widehat{\operatorname{\mathcal{E}}}$. By virtue of Theorem 7.2.3.1, it will suffice to show that the $\infty $-category $\operatorname{\mathcal{A}}= \{ \operatorname{id}_{C} \} \times _{ \operatorname{\mathcal{C}}_{/C} } ( \operatorname{\mathcal{E}}_{ / C} )_{\widetilde{Y}/}$ is weakly contractible. Since $\widehat{U}$ is an inner fibration, the map

is a right fibration (Proposition 4.3.6.8). Using our assumption that $U$ is an isofibration, we deduce that the map $\operatorname{\mathcal{E}}_{ / X } \rightarrow \operatorname{\mathcal{C}}_{/C}$ is also an isofibration. Let $\widetilde{D}$ denote the image of $\widetilde{Y}$ under this isofibration, which we identify with a morphism $f: D \rightarrow C$ in the $\infty $-category $\operatorname{\mathcal{C}}$ (so that $D = U(Y)$ and $f = \widehat{U}(v)$). Set $\operatorname{\mathcal{B}}= \{ \operatorname{id}_{C} \} \times _{ \operatorname{\mathcal{C}}_{/C} } ( \operatorname{\mathcal{C}}_{/C} )_{\widetilde{D} /}$, so that $U$ induces an isofibration

Since $\operatorname{id}_{C}$ is final when viewed as an object of $\operatorname{\mathcal{C}}_{/C}$ (Proposition 4.6.7.22), the simplicial set $\operatorname{\mathcal{B}}$ is a contractible Kan complex. Let $B \in \operatorname{\mathcal{B}}$ be the vertex which corresponds to the degenerate $2$-simplex of $\operatorname{\mathcal{C}}$ depicted in the diagram

so that the inclusion map $\{ B\} \hookrightarrow \operatorname{\mathcal{B}}$ is an equivalence of $\infty $-categories. Applying Corollary 4.5.2.30, we deduce that the inclusion map $\{ B\} \times _{\operatorname{\mathcal{B}}} \operatorname{\mathcal{A}}\hookrightarrow \operatorname{\mathcal{A}}$ is an equivalence of $\infty $-categories. In particular, it is a weak homotopy equivalence. Consequently, to show that $\operatorname{\mathcal{A}}$ is weakly contractible, it will suffice to show that the fiber $\{ B\} \times _{\operatorname{\mathcal{B}}} \operatorname{\mathcal{A}}$ is weakly contractible. Replacing $\operatorname{\mathcal{E}}$ and $\widehat{\operatorname{\mathcal{E}}}$ by the fiber products $\Delta ^1 \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}$ and $\Delta ^1 \times _{\operatorname{\mathcal{C}}} \widehat{\operatorname{\mathcal{E}}}$, respectively, we are reduced to proving that $\operatorname{\mathcal{A}}$ is weakly contractible under the additional assumption that $\operatorname{\mathcal{C}}= \Delta ^1$, where $U(Y) = 0$ and $\widehat{U}(X) = 1$.

For $i \in \{ 0,1\} $, set $\operatorname{\mathcal{E}}_{i} = \{ i\} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}$ and $\widehat{\operatorname{\mathcal{E}}}_{i} = \{ i\} \times _{\operatorname{\mathcal{C}}} \widehat{\operatorname{\mathcal{E}}}$. 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. Then the projection map $\widehat{\operatorname{\mathcal{E}}}_{ H(Y) / } \rightarrow \widehat{\operatorname{\mathcal{E}}}$ admits a covariant transport representation $\mathscr {F}: \widehat{\operatorname{\mathcal{E}}} \rightarrow \operatorname{\mathcal{S}}^{< \kappa }$, given informally by the formula $\mathscr {F}(E) = \operatorname{Hom}_{ \widehat{\operatorname{\mathcal{E}}} }( H(Y), E )$. Our assumption that $H$ exhibits $\widehat{\operatorname{\mathcal{E}}}$ as a fiberwise $\mathbb {K}$-cocompletion of $\operatorname{\mathcal{E}}$ guarantees that the restriction $\mathscr {F}|_{ \widehat{\operatorname{\mathcal{E}}}_{1} }$ preserves $K$-indexed colimits, for each $K \in \mathbb {K}$. Applying Corollary 8.4.5.10 (and our assumption that $H$ is fully faithful), we conclude that $H$ induces a left cofinal functor

By virtue of Theorem 7.2.3.1, this is a reformulation of the assertion that the $\infty $-category $\operatorname{\mathcal{A}}$ is weakly contractible (for every choice of object $X \in \widehat{\operatorname{\mathcal{E}}}_{1}$). $\square$