Definition 4.35.1. Let $p : \mathcal{S} \to \mathcal{C}$ be a functor. We say that $\mathcal{S}$ is *fibred in groupoids* over $\mathcal{C}$ if the following two conditions hold:

For every morphism $f : V \to U$ in $\mathcal{C}$ and every lift $x$ of $U$ there is a lift $\phi : y \to x$ of $f$ with target $x$.

For every pair of morphisms $\phi : y \to x$ and $ \psi : z \to x$ and any morphism $f : p(z) \to p(y)$ such that $p(\phi ) \circ f = p(\psi )$ there exists a unique lift $\chi : z \to y$ of $f$ such that $\phi \circ \chi = \psi $.

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