# Kerodon

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Remark 5.2.5.6 (Base Change). Suppose we are given a pullback diagram of simplicial sets

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

where $U$ and $U'$ are cocartesian fibrations. Then the homotopy transport representation $\operatorname{hTr}_{\operatorname{\mathcal{E}}'/\operatorname{\mathcal{C}}'}$ is isomorphic to the composite functor

$\mathrm{h} \mathit{\operatorname{\mathcal{C}}'} \rightarrow \mathrm{h} \mathit{\operatorname{\mathcal{C}}} \xrightarrow { \operatorname{hTr}_{ \operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}} } \mathrm{h} \mathit{\operatorname{QCat}}.$