Notation 9.2.8.1. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $S_{L}$ and $S_{R}$ be collections of morphisms of $\operatorname{\mathcal{C}}$. We let $\operatorname{Fun}_{L}(\Delta ^2, \operatorname{\mathcal{C}})$ denote the full subcategory of $\operatorname{Fun}(\Delta ^2, \operatorname{\mathcal{C}})$ spanned by those diagrams
where $f$ belongs to $S_{L}$, and $\operatorname{Fun}_{R}( \Delta ^2, \operatorname{\mathcal{C}})$ the full subcategory of $\operatorname{Fun}(\Delta ^2, \operatorname{\mathcal{C}})$ spanned by those diagrams where $g$ belongs to $S_{R}$. We let $\operatorname{Fun}_{LR}( \Delta ^2, \operatorname{\mathcal{C}})$ denote the intersection $\operatorname{Fun}_{L}( \Delta ^2, \operatorname{\mathcal{C}}) \cap \operatorname{Fun}_{R}( \Delta ^2, \operatorname{\mathcal{C}})$.