Kerodon

Proof. We will show that $S_{L}$ is closed under retracts; the analogous statement for $S_{R}$ follows by a similar argument. By virtue of Proposition 9.2.9.6, the restriction map

is a trivial Kan fibration. It therefore admits a section $\operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{C}}) \rightarrow \operatorname{Fun}_{LR}( \Delta ^2, \operatorname{\mathcal{C}})$, which carries each morphism $f: X \rightarrow Z$ of $\operatorname{\mathcal{C}}$ to a $2$-simplex $\sigma _{f}:$

where $f_{L} \in S_{L}$ and $f_{R} \in S_{R}$. We will complete the proof by showing that $f$ belongs to $S_{L}$ if and only if $f_{R}$ is an isomorphism in $\operatorname{\mathcal{C}}$. One direction is clear: if $f_{R}$ is an isomorphism, then $f$ is isomorphic to $f_{L}$ in the $\infty $-category $\operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{C}})$, and therefore belongs to $S_{L}$ by virtue of our assumption that $S_ L$ is closed under isomorphism. For the converse, assume that $f$ belongs to $S_{L}$. Since $\operatorname{id}_{Z}$ belongs to $S_{R}$ (Corollary 9.2.9.7), the degenerate $2$-simplex $\widetilde{f} = s^{1}_{1}(f)$ can be regarded as an object of $\operatorname{Fun}_{LR}( \Delta ^2, \operatorname{\mathcal{C}})$ satisfying $D( \widetilde{f} ) = f = D( \sigma _{f} )$. Since $D$ is an equivalence of $\infty $-categories, the $2$-simplex $\sigma _{f}$ is isomorphic to $\widetilde{f}$ as an object of the $\infty $-category $\operatorname{Fun}_{LR}( \Delta ^2, \operatorname{\mathcal{C}})$. It follows that $f_{R} = d^{2}_{0}( \sigma _{f} )$ is isomorphic to $\operatorname{id}_{Z} = d^{2}_{0}( \widetilde{f} )$ as an object of the $\infty $-category $\operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{C}})$, so that $f_{R}$ is an isomorphism (Example 4.4.1.14). $\square$