Projection from Product Category

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Theorem

Let $\mathcal C$ and $\mathcal D$ be categories.

Let $\mathcal C \times \mathcal D$ be the product category.

The projections:

$\pi_{\mathcal C} : \mathcal C \times \mathcal D \to \mathcal C : (f,g) \mapsto f$
$\pi_{\mathcal D} : \mathcal C \times \mathcal D \to \mathcal D : (f,g) \mapsto g$

where $(f,g) \in \operatorname{ob}(\mathcal C \times \mathcal D)$ or $(f,g)\in \operatorname{mor}(\mathcal C \times \mathcal D)$ are functors.

Moreover, $\pi_{\mathcal C}$ and $\pi_{\mathcal D}$ satisfy the following universal property:

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

That is, the following diagram commutes:

Product Category Universal Property.png

Proof