Kerodon

Proof. Since $\mathscr {E}$ is isofibrant, Proposition 4.5.6.14 guarantees that the functor

is an equivalence of $\infty $-categories. In particular, there exists a natural transformation $\beta : \mathscr {F} \rightarrow \mathscr {E}$ such that $\beta \circ \alpha $ is isomorphic to $\operatorname{id}_{ \mathscr {E} }$ (when viewed as an object of the $\infty $-category $\operatorname{Hom}_{ \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{Set_{\Delta }})} ( \mathscr {E}, \mathscr {E} )_{\bullet }$). To complete the proof, it will suffice to show that $\beta $ is also a right homotopy inverse to $\alpha $: that is, the composition $\alpha \circ \beta $ is isomorphic to $\operatorname{id}_{ \mathscr {F}}$ (when viewed as an object of the $\infty $-category $\operatorname{Hom}_{ \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{Set_{\Delta }})} ( \mathscr {F}, \mathscr {F} )_{\bullet }$).

For each object $C \in \operatorname{\mathcal{C}}$, the functor $\beta _{C}: \mathscr {F}(C) \rightarrow \mathscr {E}(C)$ is a left homotopy inverse of the functor $\alpha _{C}: \mathscr {E}(C) \rightarrow \mathscr {F}(C)$. Since $\alpha _ C$ is an equivalence of $\infty $-categories, it follows that $\beta _{C}$ is also an equivalence of $\infty $-categories. Allowing $C$ to vary, we conclude that $\beta $ is a levelwise categorical equivalence. We can therefore repeat the preceding argument to obtain a natural transformation $\gamma : \mathscr {E} \rightarrow \mathscr {F}$ such that $\gamma \circ \beta $ is isomorphic to $\operatorname{id}_{ \mathscr {F} }$. We then have isomorphisms

in the $\infty $-category $\operatorname{Hom}_{ \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{Set_{\Delta }})} ( \mathscr {E}, \mathscr {F} )_{\bullet }$, so that $\alpha \circ \beta $ is also isomorphic to $\operatorname{id}_{ \mathscr {F}}$. $\square$