\begin{definition}[Definition:Domain Functor]
Let $\mathbf C$ be a metacategory.
Let $\mathbf C^\to$ be its morphism category.
The '''domain functor''' is the functor $\operatorname{\mathbf{dom}}: \mathbf C^\to \to \mathbf C$ defined by:
{{begin-axiom}}
{{axiom|lc= Object functor:
       |m = \operatorname{\mathbf{dom} } f := \operatorname{dom} f
}}
{{axiom|lc= Morphism functor:
       |m = \operatorname{\mathbf{dom} } \left({g_1, g_2}\right) := g_1
}}
{{end-axiom}}
That it is in fact a functor is shown on Domain Functor is Functor.
The functor $\mathbf{dom}$ can be represented as follows:
::$\begin{xy}
<0em,2em>*+{A} = "A",
<0em,-2em>*+{B} = "B",
<4em,2em>*+{A'} = "A2",
<4em,-2em>*+{B'} = "B2",
"A";"B" **@{-} ?>*@{>} ?*!/^1em/{f},
"A";"A2" **@{-} ?>*@{>} ?*!/_1em/{g_1},
"A2";"B2" **@{-} ?>*@{>} ?*!/_1em/{f'},
"B";"B2" **@{-} ?>*@{>} ?*!/^1em/{g_2},
<6em,0em>;<10em,0em> **@{~} ?>*@2{>} ?*!/_1em/{\mathbf{dom}},
<12em,2em>*+{A} = "AA",
<16em,2em>*+{A'} = "AA2",
"AA";"AA2" **@{-} ?>*@{>} ?*!/_1em/{g_1},
\end{xy}$
It is thus seen to be an example of a forgetful functor.
\end{definition}