Lemma 4.41.3. In the situation above the fibre category of $(\mathcal{C}/U) \times _\mathcal {Y} \mathcal{X}$ over an object $f : V \to U$ of $\mathcal{C}/U$ is the category described as follows:

1. objects are pairs $(x, \phi )$, where $x \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{X}_ V)$, and $\phi : f^*y \to F(x)$ is a morphism in $\mathcal{Y}_ V$,

2. the set of morphisms between $(x, \phi )$ and $(x', \phi ')$ is the set of morphisms $\psi : x \to x'$ in $\mathcal{X}_ V$ such that $F(\psi ) = \phi ' \circ \phi ^{-1}$.

Proof. See discussion above. $\square$

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