Definition 4.31.2. Let $\mathcal{C}$ be a $(2, 1)$-category. Let $x, y, z\in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$ and $f\in \mathop{\mathrm{Mor}}\nolimits _\mathcal {C}(x, z)$ and $g\in \mathop{\mathrm{Mor}}\nolimits _{\mathcal C}(y, z)$. A 2-fibre product of $f$ and $g$ is a final object in the category of 2-commutative diagrams described above. If a 2-fibre product exists we will denote it $x \times _ z y\in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$, and denote the required morphisms $p\in \mathop{\mathrm{Mor}}\nolimits _{\mathcal C}(x \times _ z y, x)$ and $q\in \mathop{\mathrm{Mor}}\nolimits _{\mathcal C}(x \times _ z y, y)$ making the diagram

$\xymatrix{ & x \times _ z y \ar[r]^{p} \ar[d]_ q & x \ar[d]^{f} \\ & y \ar[r]^{g} & z }$

2-commute and we will denote the given invertible 2-morphism exhibiting this by $\psi : f \circ p \to g \circ q$.

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