The Stacks project

Lemma 4.31.6. Let

\[ \xymatrix{ & \mathcal{Y} \ar[d]_ I \ar[rd]^ K & \\ \mathcal{X} \ar[r]^ H \ar[rd]^ L & \mathcal{Z} \ar[rd]^ M & \mathcal{B} \ar[d]^ G \\ & \mathcal{A} \ar[r]^ F & \mathcal{C} } \]

be a $2$-commutative diagram of categories. A choice of isomorphisms $\alpha : G \circ K \to M \circ I$ and $\beta : M \circ H \to F \circ L$ determines a morphism

\[ \mathcal{X} \times _\mathcal {Z} \mathcal{Y} \longrightarrow \mathcal{A} \times _\mathcal {C} \mathcal{B} \]

of $2$-fibre products associated to this situation.

Proof. Just use the functor

\[ (X, Y, \phi ) \longmapsto (L(X), K(Y), \alpha ^{-1}_ Y \circ M(\phi ) \circ \beta ^{-1}_ X) \]

on objects and

\[ (a, b) \longmapsto (L(a), K(b)) \]

on morphisms. $\square$

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