\begin{definition}[Definition:Morphism Category]
Let $\mathbf C$ be a metacategory.
Its '''morphism category''', denoted $\mathbf C^\to$, is defined as follows:
{{DefineCategory
|ob = The morphisms $\mathbf C_1$ of $\mathbf C$
|mor = $g: f \to f'$ is a pair $\left({g_1, g_2}\right)$ of morphisms of $\mathbf C$ such that $g_2 \circ f = f' \circ g_1$
|comp = $\left({h_1, h_2}\right) \circ \left({g_1, g_2}\right) := \left({h_1 \circ g_1, h_2 \circ g_2}\right)$, whenever this is defined
|id = $\operatorname{id}_{f} := \left({\operatorname{id}_C, \operatorname{id}_D}\right)$ for $f: C \to D$
}}
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:
::$\begin{xy}
<4em,5em>*{\left({h_1, h_2}\right) \circ \left({g_1, g_2}\right)},
<0em,2em>*+{A} = "A",
<0em,-2em>*+{B} = "B",
<4em,2em>*+{A'} = "A2",
<4em,-2em>*+{B'} = "B2",
"A";"B" **@{-} ?>*@{>} ?*!/^1em/{f},
"A";"A2" **@{-} ?>*@{>} ?*!/_1em/{g_1},
"A2";"B2" **@{-} ?>*@{>} ?*!/_1em/{f'},
"B";"B2" **@{-} ?>*@{>} ?*!/^1em/{g_2},
<8em,2em>*+{A''} = "A3",
<8em,-2em>*+{B''} = "B3",
"A2";"A3" **@{-} ?>*@{>} ?*!/_1em/{h_1},
"B2";"B3" **@{-} ?>*@{>} ?*!/^1em/{h_2},
"A3";"B3" **@{-} ?>*@{>} ?*!/_1em/{f''},
<12em,5em>*{=},
<10em,0em>;<14em,0em> **@{~} ?>*@2{>},
<20em,5em>*+{\left({h_1 \circ g_1, h_2 \circ g_2}\right)},
<16em,2em>*+{A} = "AA",
<16em,-2em>*+{B} = "BB",
<24em,2em>*+{A''} = "AA3",
<24em,-2em>*+{B''} = "BB3",
"AA";"BB" **@{-} ?>*@{>} ?*!/^1em/{f},
"AA";"AA3" **@{-} ?>*@{>} ?*!/_1em/{h_1 \circ g_1},
"AA3";"BB3" **@{-} ?>*@{>} ?*!/_1em/{f''},
"BB";"BB3" **@{-} ?>*@{>} ?*!/^1em/{h_2 \circ g_2},
\end{xy}$
\end{definition}