$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proposition 9.6.6.10 (Transitivity). Let $\operatorname{\mathcal{C}}$ be an $\infty $-category containing a morphism $g: X \rightarrow Y$ and a $2$-simplex
9.33
\begin{equation} \begin{gathered}\label{equation:two-out-of-three-for-orthogonal} \xymatrix@R =50pt@C=50pt{ & B \ar [dr]^{ f'' } & \\ A \ar [ur]^{f'} \ar [rr]^{f} & & C. } \end{gathered} \end{equation}
Assume that $f'$ is left orthogonal to $g$. Then $f$ is left orthogonal to $g$ if and only if $f''$ is left orthogonal to $g$.
Proof.
We have a commutative diagram of morphism spaces
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{Hom}_{\operatorname{\mathcal{C}}}(C,X) \ar [r] \ar [d] & \operatorname{Hom}_{\operatorname{\mathcal{C}}}(B,X) \ar [r] \ar [d] & \operatorname{Hom}_{\operatorname{\mathcal{C}}}(A,X) \ar [d] \\ \operatorname{Hom}_{\operatorname{\mathcal{C}}}(C, Y) \ar [r] & \operatorname{Hom}_{\operatorname{\mathcal{C}}}( B, Y) \ar [r] & \operatorname{Hom}_{\operatorname{\mathcal{C}}}(A, Y) } \]
where the right square is a pullback. Applying Proposition 7.6.2.40, we see that the left square is a pullback if and only if the outer rectangle is a pullback.
$\square$