Definition:Projection Functor

Definition
Let $\mathbf C$ and $\mathbf D$ be metacategories, and let $\mathbf C \times \mathbf D$ be their product.

The first projection functor $\operatorname{pr}_1: \mathbf C \times \mathbf D \to \mathbf C$ is defined by:


 * $\operatorname{pr}_1 \left({C, D}\right) := C$ for all objects $\left({C, D}\right) \in \operatorname{ob} \mathbf C \times \mathbf D$
 * $\operatorname{pr}_1 \left({f, g}\right) := f$ for all morphisms $\left({f, g}\right) \in \operatorname{mor} \mathbf C \times \mathbf D$

The second projection functor $\operatorname{pr}_2: \mathbf C \times \mathbf D \to \mathbf D$ is defined by:


 * $\operatorname{pr}_2 \left({C, D}\right) := D$ for all objects $\left({C, D}\right) \in \operatorname{ob} \mathbf C \times \mathbf D$
 * $\operatorname{pr}_2 \left({f, g}\right) := g$ for all morphisms $\left({f, g}\right) \in \operatorname{mor} \mathbf C \times \mathbf D$

That these constitute functors is shown on Projection Functor is Functor.