Projection from Product Category

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