# Kerodon

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Variant 4.2.2.4. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a functor between categories. We say that $U$ is a opfibration in groupoids if the following conditions are satisfied:

$(A')$

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

$(B')$

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

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

is a pullback square.

In this case, we will also say that $\operatorname{\mathcal{E}}$ is opfibered in groupoids over $\operatorname{\mathcal{C}}$.