Kerodon

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Definition 4.5.9.10. Let $U: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$ be a morphism of simplicial sets. We will say that $U$ is exponentiable if it satisfies the following condition:

$(\ast )$

For every diagram of simplicial sets

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{D}}'' \ar [r]^-{F} \ar [d] & \operatorname{\mathcal{D}}' \ar [r] \ar [d] & \operatorname{\mathcal{D}}\ar [d]^{U} \\ \operatorname{\mathcal{C}}'' \ar [r]^-{ \overline{F} } & \operatorname{\mathcal{C}}' \ar [r] & \operatorname{\mathcal{C}}} \]

in which both squares are pullbacks, if $\overline{F}$ is a categorical equivalence, then $F$ is also a categorical equivalence.