Definition:Codomain Functor

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Let $\mathbf C$ be a metacategory.

Let $\mathbf C^\to$ be its morphism category.

The codomain functor is the functor $\operatorname{\mathbf{cod}}: \mathbf C^\to \to \mathbf C$ defined by:

Object functor:    \(\ds \operatorname{\mathbf{cod} } f := \operatorname{cod} f \)      
Morphism functor:    \(\ds \operatorname{\mathbf{cod} } \left({g_1, g_2}\right) := g_2 \)      

That it is in fact a functor is shown on Codomain Functor is Functor.

The functor $\mathbf{cod}$ can be represented as follows:

$\begin{xy} <0em,2em>*+{A} = "A", <0em,-2em>*+{B} = "B", <4em,2em>*+{A'} = "A2", <4em,-2em>*+{B'} = "B2", "A";"B" **@{-} ?>*@{>} ?*!/^1em/{f}, "A";"A2" **@{-} ?>*@{>} ?*!/_1em/{g_1}, "A2";"B2" **@{-} ?>*@{>} ?*!/_1em/{f'}, "B";"B2" **@{-} ?>*@{>} ?*!/^1em/{g_2}, <6em,0em>;<10em,0em> **@{~} ?>*@2{>} ?*!/_1em/{\mathbf{cod}}, <12em,-2em>*+{B} = "BB", <16em,-2em>*+{B'} = "BB2", "BB";"BB2" **@{-} ?>*@{>} ?*!/^1em/{g_2}, \end{xy}$

It is thus seen to be an example of a forgetful functor.