Kerodon

$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$

Remark 4.2.5.8. Suppose we are given a pullback diagram

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

in the ordinary category $\operatorname{Cat}$ (so that the category $\operatorname{\mathcal{C}}'$ is isomorphic to the fiber product $\operatorname{\mathcal{C}}\times _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}'$). If $F$ is a fibration in groupoids, then so is $F'$. Similarly, if $F$ is an opfibration in groupoids, then so is $F'$.