Characterization of Metacategory via Equations

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

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

Let $\operatorname{id}$ associate 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 $\circ$ associate 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.