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Remark 4.5.9.7. Suppose we are given a commutative diagram of simplicial sets

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{E}}' \ar [r] \ar [d] & \operatorname{\mathcal{D}}' \ar [r] \ar [d] & \operatorname{\mathcal{C}}' \ar [d] \\ \operatorname{\mathcal{E}}\ar [r] & \operatorname{\mathcal{D}}\ar [r] & \operatorname{\mathcal{C}}, } \]

where both squares are pullbacks. Then there is a canonical isomorphism of simplicial sets

\[ \operatorname{Res}_{ \operatorname{\mathcal{D}}' / \operatorname{\mathcal{C}}' }( \operatorname{\mathcal{E}}' ) \simeq \operatorname{\mathcal{C}}' \times _{\operatorname{\mathcal{C}}} \operatorname{Res}_{\operatorname{\mathcal{D}}/\operatorname{\mathcal{C}}}( \operatorname{\mathcal{E}}). \]