# Kerodon

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Remark 5.2.3.9 (Base Change). Suppose we are given a commutative diagram of simplicial sets

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

where both squares are pullbacks. Then the induced diagram

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

is also a pullback square.