Remark 4.2.2.8. Suppose we are given a pullback diagram
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{E}}' \ar [r] \ar [d]^{U'} & \operatorname{\mathcal{E}}\ar [d]^{U} \\ \operatorname{\mathcal{C}}' \ar [r] & \operatorname{\mathcal{C}}} \]
in the ordinary category $\operatorname{Cat}$ (so that the category $\operatorname{\mathcal{E}}'$ is isomorphic to the fiber product $\operatorname{\mathcal{C}}' \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}$). If $U$ is a fibration in groupoids, then so is $U'$. Similarly, if $U$ is an opfibration in groupoids, then so is $U'$.