Kerodon

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Corollary 6.3.5.5. Suppose we are given a pullback diagram of simplicial sets

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

where $U$ and $U'$ are left fibrations. Suppose that $\overline{F}$ exhibits $\operatorname{\mathcal{C}}'$ as a localization of $\operatorname{\mathcal{C}}$ with respect to some collection of edges $\overline{W}$. Then $F$ exhibits $\operatorname{\mathcal{E}}'$ as a localization of $\operatorname{\mathcal{E}}$ with respect to $W = U^{-1}( \overline{W} )$.