Definition:Morphism Category

Definition
Let $\mathbf C$ be a metacategory.

Its morphism category, denoted $\mathbf C^\to$, is defined as follows:

That $\mathbf C^\to$ is a metacategory is shown on Morphism Category is Category.

The morphisms of $\mathbf C^\to$ can be made more intuitive by the following diagram:


 * $\begin{xy}

<-2em,0em>*+{f} = "f", <2em,0em>*+{f'} = "f2",

"f";"f2" **@{-} ?>*@{>} ?*!/_1em/{\scriptstyle \left({g_1, g_2}\right)},

<3em,0em>*{:},

<7em,2em>*+{A} = "A", <7em,-2em>*+{B} = "B", <11em,2em>*+{A'} = "A2", <11em,-2em>*+{B'} = "B2",

"A";"B" **@{-} ?>*@{>} ?*!/^1em/{f}, "A";"A2" **@{-} ?>*@{>} ?*!/_1em/{g_1}, "A2";"B2" **@{-} ?>*@{>} ?*!/_1em/{f'}, "B";"B2" **@{-} ?>*@{>} ?*!/^1em/{g_2} \end{xy}$

The composition likewise benefits from a diagrammatic representation:


 * $\begin{xy}

<4em,5em>*{\left({h_1, h_2}\right) \circ \left({g_1, g_2}\right)},

<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},

<8em,2em>*+{A''} = "A3", <8em,-2em>*+{B''} = "B3",

"A2";"A3" **@{-} ?>*@{>} ?*!/_1em/{h_1}, "B2";"B3" **@{-} ?>*@{>} ?*!/^1em/{h_2}, "A3";"B3" **@{-} ?>*@{>} ?*!/_1em/{f''},

<12em,5em>*{=}, <10em,0em>;<14em,0em> **@{~} ?>*@2{>},

<20em,5em>*+{\left({h_1 \circ g_1, h_2 \circ g_2}\right)},

<16em,2em>*+{A} = "AA", <16em,-2em>*+{B} = "BB", <24em,2em>*+{A''} = "AA3", <24em,-2em>*+{B''} = "BB3",

"AA";"BB" **@{-} ?>*@{>} ?*!/^1em/{f}, "AA";"AA3" **@{-} ?>*@{>} ?*!/_1em/{h_1 \circ g_1}, "AA3";"BB3" **@{-} ?>*@{>} ?*!/_1em/{f''}, "BB";"BB3" **@{-} ?>*@{>} ?*!/^1em/{h_2 \circ g_2}, \end{xy}$

Also known as
The morphism category $\mathbf C^\to$ is also called the arrow category.