Kerodon

Proof. Let $W$ be the collection of all isomorphisms in $\operatorname{\mathcal{C}}$, and set $S_{L}^{+} = S_{L} \cup W$ and $S_{R}^{+} = S_{R} \cup W$. Using Corollary 9.2.7.14, we deduce that $S_{L}^{+}$ is left orthogonal to $S_{R}^{+}$, so that $(S_{L}^{+}, S_{R}^{+} )$ is also a factorization system on $\operatorname{\mathcal{C}}$. Let $\operatorname{Fun}_{LR}(\Delta ^2, \operatorname{\mathcal{C}})$ be as in Notation 9.2.8.1 and define $\operatorname{Fun}_{LR}^{+}( \Delta ^2, \operatorname{\mathcal{C}})$ similarly. We then have a commutative diagram

where both of the vertical maps are trivial Kan fibrations (Proposition 9.2.9.6). It follows that the inclusion map $\operatorname{Fun}_{LR}( \Delta ^2, \operatorname{\mathcal{C}}) \hookrightarrow \operatorname{Fun}_{LR}^{+}( \Delta ^2, \operatorname{\mathcal{C}})$ is an equivalence of $\infty $-categories. Since $\operatorname{Fun}_{LR}( \Delta ^2, \operatorname{\mathcal{C}})$ is a replete full subcategory of $\operatorname{Fun}_{LR}^{+}( \Delta ^2, \operatorname{\mathcal{C}})$, we must have $\operatorname{Fun}_{LR}( \Delta ^2, \operatorname{\mathcal{C}}) = \operatorname{Fun}_{LR}^{+}( \Delta ^2, \operatorname{\mathcal{C}})$. In particular, if $f: X \rightarrow Y$ is an isomorphism in $\operatorname{\mathcal{C}}$, then the degenerate $2$-simplices $s^{1}_{0}(f)$ and $s^{1}_{1}(f)$ are both contained in $\operatorname{Fun}_{LR}( \Delta ^2, \operatorname{\mathcal{C}})$, so that $f$ is contained in both $S_{L}$ and $S_{R}$. $\square$