Projection from Product Category

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Theorem

Let $\CC, \DD$ be categories.

Let $\CC \times \DD$ be the product category.

The projections:

$\pi_\CC: \CC \times \DD \to \CC : \tuple {f, g} \mapsto f$
$\pi_\DD: \CC \times \DD \to \DD : \tuple {f, g} \mapsto g$

where $\tuple {f, g} \in \map {\operatorname{ob} } {\CC \times \DD}$ or $\tuple {f, g} \in \map {\operatorname{mor} } {\CC \times \DD}$ are functors.

Moreover, $\pi_\CC, \pi_\DD$ satisfy the following universal property:

For any category $\EE$ and any functors $F : \EE \to \CC$, $G : \mathcal E \to \mathcal D$, there exists a unique functor $H : \EE \to \CC \times \DD$ such that $F = \pi_\CC H$ and $G = \pi_\DD H$

That is, the following diagram commutes:

Product Category Universal Property.png

Proof