Characterization of Metacategory via Equations

Theorem
Let $\mathbf C_0$ and $\mathbf C_1$ be collections of objects.

Let $\operatorname{cod}$ and $\operatorname{dom}$ assign to every element of $\mathbf C_1$ an element of $\mathbf C_0$.

Let $\operatorname{id}$ assign to every element of $\mathbf C_0$ an element of $\mathbf C_1$.

Denote with $\mathbf C_2$ the collection of pairs $\left({f, g}\right)$ of elements of $\mathbf C_1$ satisfying:


 * $\operatorname{dom} g = \operatorname{cod} f$

Let $\circ$ assign to every such pair an element of $\mathbf C_1$.

Then $\mathbf C_0, \mathbf C_1, \operatorname{cod}, \operatorname{dom}, \operatorname{id}$ and $\circ$ together determine a metacategory $\mathbf C$ iff the following seven axioms are satisfied:

with $A$ and $f,g,h$ arbitrary elements of $\mathbf C_0$ and $\mathbf C_1$, respectively.

Further, in the last two lines it is presumed that all compositions are defined.

Hence it follows that:


 * $\mathbf C_0$ and $\mathbf C_1$ represent the collections of objects and morphisms of $\mathbf C$
 * $\operatorname{dom}$ and $\operatorname{cod}$ represent the domain and codomain of a morphism of $\mathbf C$
 * $\operatorname{id}$ represents the identity morphisms of $\mathbf C$
 * $\mathbf C_2$ represents the collection of composable morphisms of $\mathbf C$
 * $\circ$ represents the composition of morphisms in $\mathbf C$