$\xymatrix{ \mathcal{U} \ar[d] \ar[r] & \mathcal{V} \ar[d] \\ \mathcal{X} \ar[r] & \mathcal{Y} }$

be a $2$-fibre product of categories. Then the diagram

$\xymatrix{ \mathcal{U} \ar[d] \ar[r] & \mathcal{U} \times _\mathcal {V} \mathcal{U} \ar[d] \\ \mathcal{X} \ar[r] & \mathcal{X} \times _\mathcal {Y} \mathcal{X} }$

is $2$-cartesian.

Proof. This is a purely $2$-category theoretic statement, valid in any $(2, 1)$-category with $2$-fibre products. Explicitly, it follows from the following chain of equivalences:

\begin{align*} \mathcal{X} \times _{(\mathcal{X} \times _\mathcal {Y} \mathcal{X})} (\mathcal{U} \times _\mathcal {V} \mathcal{U}) & = \mathcal{X} \times _{(\mathcal{X} \times _\mathcal {Y} \mathcal{X})} ((\mathcal{X} \times _\mathcal {Y} \mathcal{V}) \times _\mathcal {V} (\mathcal{X} \times _\mathcal {Y} \mathcal{V})) \\ & = \mathcal{X} \times _{(\mathcal{X} \times _\mathcal {Y} \mathcal{X})} (\mathcal{X} \times _\mathcal {Y} \mathcal{X} \times _\mathcal {Y} \mathcal{V}) \\ & = \mathcal{X} \times _\mathcal {Y} \mathcal{V} = \mathcal{U} \end{align*}

see Lemmas 4.31.8 and 4.31.10. $\square$

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