Proposition 9.6.6.12. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category containing a morphism $g: X \rightarrow Y$. Then the collection of morphisms which are left orthogonal to $g$ is closed under levelwise colimits in $\operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{C}})$.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proof. For every morphism $f: A \rightarrow B$ of $\operatorname{\mathcal{C}}$, let $\sigma _{f}$ denote the diagram of morphism spaces
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{Hom}_{\operatorname{\mathcal{C}}}(B,X) \ar [r] \ar [d] & \operatorname{Hom}_{\operatorname{\mathcal{C}}}(A,X) \ar [d] \\ \operatorname{Hom}_{\operatorname{\mathcal{C}}}( B, Y) \ar [r] & \operatorname{Hom}_{\operatorname{\mathcal{C}}}(A, Y) } \]
appearing in Definition 9.6.6.1. Using Proposition 7.4.1.22, we see that the construction $f \mapsto \sigma _{f}$ carries (levelwise) colimit diagrams in $\operatorname{Fun}(\Delta ^1, \operatorname{\mathcal{C}})$ to limit diagrams in the $\infty $-category $\operatorname{Fun}( \Delta ^1 \times \Delta ^1, \operatorname{\mathcal{S}})$. Applying Corollary 7.3.8.6, we see that the collection of those morphisms $f$ such that $\sigma _{f}$ is a pullback diagram is closed under levelwise colimits in $\operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{C}})$. $\square$