Kerodon

$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$ $\newcommand\empty{}$
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$

Remark 5.1.6.3 (Two-out-of-Three). Suppose we are given a commutative diagram of simplicial sets

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{C}}\ar [dr] \ar [r]^-{F} & \operatorname{\mathcal{C}}' \ar [d] \ar [r]^-{F'} & \operatorname{\mathcal{C}}'' \ar [dl] \\ & \operatorname{\mathcal{E}}, & } \]

where the vertical maps are inner fibrations. If any two of the morphisms $F$, $F'$, and $F' \circ F$ are equivalences of inner fibrations over $\operatorname{\mathcal{E}}$, then so is the third. In particular, the collection of equivalences of inner fibrations over $\operatorname{\mathcal{E}}$ is closed under composition.