The Stacks project

Lemma 4.31.8. Let

\[ \xymatrix{ \mathcal{A} \ar[rd] & & \mathcal{C} \ar[ld] \ar[rd] & & \mathcal{E} \ar[ld] \\ & \mathcal{B} & & \mathcal{D} } \]

be a diagram of categories and functors. Then there is a canonical isomorphism

\[ (\mathcal{A} \times _\mathcal {B} \mathcal{C}) \times _\mathcal {D} \mathcal{E} \cong \mathcal{A} \times _\mathcal {B} (\mathcal{C} \times _\mathcal {D} \mathcal{E}) \]

of categories.

Proof. Just use the functor

\[ ((A, C, \phi ), E, \psi ) \longmapsto (A, (C, E, \psi ), \phi ) \]

if you know what I mean. $\square$

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