Kerodon

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

Variant 4.2.5.5. Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be a categories, and let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor. We say that $F$ is an opfibration in groupoids if the opposite functor $F^{\operatorname{op}}: \operatorname{\mathcal{C}}^{\operatorname{op}} \rightarrow \operatorname{\mathcal{D}}^{\operatorname{op}}$ is a fibration in groupoids. In other words, $F$ is an opfibration in groupoids if and only if it satisfies the following conditions:

$(A')$

For every object $X \in \operatorname{\mathcal{C}}$ and every morphism $\overline{u}: F(X) \rightarrow \overline{Y}$ in $\operatorname{\mathcal{D}}$, there exists a morphism $u: X \rightarrow Y$ in $\operatorname{\mathcal{C}}$ with $\overline{Y} = F(Y)$ and $\overline{u} = F(u)$.

$(B')$

For every morphism $u: X \rightarrow Y$ in $\operatorname{\mathcal{C}}$ and every object $Z \in \operatorname{\mathcal{C}}$, the diagram of sets

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{Hom}_{\operatorname{\mathcal{C}}}(Y,Z) \ar [r]^-{\circ u} \ar [d]^{F} & \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Z) \ar [d]^{F} \\ \operatorname{Hom}_{\operatorname{\mathcal{D}}}( F(Y), F(Z) ) \ar [r]^-{ \circ F(u)} & \operatorname{Hom}_{\operatorname{\mathcal{D}}}( F(X), F(Z) ) } \]

is a pullback square.