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

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. $\square$

There are also:

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