\begin{definition}[Definition:Projection Functor]
Let $\mathbf C$ and $\mathbf D$ be metacategories, and let $\mathbf C \times \mathbf D$ be their product.
The '''first projection functor''' $\pr_1: \mathbf C \times \mathbf D \to \mathbf C$ is defined by:
:$\map {\pr_1} {C, D} := C$ for all objects $\tuple {C, D} \in \operatorname{ob} \mathbf C \times \mathbf D$
:$\map {\pr_1} {f, g} := f$ for all morphisms $\tuple {f, g} \in \operatorname{mor} \mathbf C \times \mathbf D$
The '''second projection functor''' $\pr_2: \mathbf C \times \mathbf D \to \mathbf D$ is defined by:
:$\map {\pr_2} {C, D} := D$ for all objects $\tuple {C, D} \in \operatorname{ob} \mathbf C \times \mathbf D$
:$\map {\pr_2} {f, g} := g$ for all morphisms $\tuple {f, g} \in \operatorname{mor} \mathbf C \times \mathbf D$
That these constitute functors is shown on Projection Functor is Functor.
\end{definition}