Projection from Product Category

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Let $\CC, \DD$ be categories.

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

Then the projection functors:

$\pi_\CC: \CC \times \DD \to \CC$
$\pi_\DD: \CC \times \DD \to \DD$

are indeed functors.

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

For any category $\EE$ and any functors $F : \EE \to \CC$, $G : \EE \to \DD$, 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:

$\xymatrix { \CC & \CC \times \DD \ar[l]_{\pi_\CC} \ar[r]^{\pi_\DD} & \DD \\ & \EE \ar[lu]^F \ar[u]_{\exists ! H} \ar[ru]_G}$