# 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:

 $\displaystyle \operatorname{dom} \operatorname{id}_A = A$ $\qquad$ $\displaystyle \operatorname{cod} \operatorname{id}_A = A$ $\displaystyle f \circ \operatorname{id}_{\operatorname{dom} f} = f$  $\displaystyle \operatorname{id}_{\operatorname{cod} f} \circ f = f$ $\displaystyle \operatorname{dom} \left({g \circ f}\right) = \operatorname{dom} f$  $\displaystyle \operatorname{cod} \left({g \circ f}\right) = \operatorname{cod} g$ $\displaystyle h \circ \left({g \circ f}\right)$ $=$ $\displaystyle \left({h \circ g}\right) \circ f$

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: