Kerodon

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

Remark 4.5.9.14. The collection of exponentiable morphisms of simplicial sets is closed under retracts. That is, if we are given a commutative diagram of simplicial sets

\[ \xymatrix@C =40pt@R=40pt{ \operatorname{\mathcal{D}}\ar [r] \ar [d]^{U} & \operatorname{\mathcal{D}}' \ar [d]^{U'} \ar [r] & \operatorname{\mathcal{D}}\ar [d]^{U} \\ \operatorname{\mathcal{C}}\ar [r] & \operatorname{\mathcal{C}}' \ar [r] & \operatorname{\mathcal{C}}} \]

where $U'$ is exponentiable and both horizontal compositions are the identity, then $U$ is also exponentiable.