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

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

$\text{pr}_{02} : \mathcal{A} \times _\mathcal {B} \mathcal{C} \times _\mathcal {D} \mathcal{E} \longrightarrow \mathcal{A} \times _\mathcal {F} \mathcal{E}$

of categories.

Proof. If we write $\mathcal{A} \times _\mathcal {B} \mathcal{C} \times _\mathcal {D} \mathcal{E}$ as $(\mathcal{A} \times _\mathcal {B} \mathcal{C}) \times _\mathcal {D} \mathcal{E}$ then we can just use the functor

$((A, C, \phi ), E, \psi ) \longmapsto (A, E, G(\psi ) \circ F(\phi ))$

if you know what I mean. $\square$

There are also:

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