Lemma 4.32.3. Let $\mathcal{C}$ be a category. The $(2, 1)$-category of categories over $\mathcal{C}$ has 2-fibre products. Suppose that $F : \mathcal{X} \to \mathcal{S}$ and $G : \mathcal{Y} \to \mathcal{S}$ are morphisms of categories over $\mathcal{C}$. An explicit 2-fibre product $\mathcal{X} \times _\mathcal {S}\mathcal{Y}$ is given by the following description

an object of $\mathcal{X} \times _\mathcal {S} \mathcal{Y}$ is a quadruple $(U, x, y, f)$, where $U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$, $x\in \mathop{\mathrm{Ob}}\nolimits (\mathcal{X}_ U)$, $y\in \mathop{\mathrm{Ob}}\nolimits (\mathcal{Y}_ U)$, and $f : F(x) \to G(y)$ is an isomorphism in $\mathcal{S}_ U$,

a morphism $(U, x, y, f) \to (U', x', y', f')$ is given by a pair $(a, b)$, where $a : x \to x'$ is a morphism in $\mathcal{X}$, and $b : y \to y'$ is a morphism in $\mathcal{Y}$ such that

$a$ and $b$ induce the same morphism $U \to U'$, and

the diagram

\[ \xymatrix{ F(x) \ar[r]^ f \ar[d]^{F(a)} & G(y) \ar[d]^{G(b)} \\ F(x') \ar[r]^{f'} & G(y') } \]is commutative.

The functors $p : \mathcal{X} \times _\mathcal {S}\mathcal{Y} \to \mathcal{X}$ and $q : \mathcal{X} \times _\mathcal {S}\mathcal{Y} \to \mathcal{Y}$ are the forgetful functors in this case. The transformation $\psi : F \circ p \to G \circ q$ is given on the object $\xi = (U, x, y, f)$ by $\psi _\xi = f : F(p(\xi )) = F(x) \to G(y) = G(q(\xi ))$.

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