Kerodon

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

Definition 4.4.1.1. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor between categories. We say that $F$ is an isofibration if it satisfies the following condition:

$(\ast )$

For every object $C \in \operatorname{\mathcal{C}}$ and every isomorphism $u: D \rightarrow F(C)$ in the category $\operatorname{\mathcal{D}}$, there exists an isomorphism $\overline{u}: \overline{D} \rightarrow C$ in the category $\operatorname{\mathcal{C}}$ satisfying $F(\overline{u}) = u$.