# Category Axioms are Self-Dual

## Theorem

### Morphisms-Only Category Theory

Let $\mathrm{MOCT}$ be the collection of axioms for morphisms-only category theory.

Then:

$\mathrm{MOCT} = \mathrm{MOCT}^*$

where $\mathrm{MOCT}^*$ consists of the dual statements of those in $\mathrm{MOCT}$.

### Object Category Theory

Let $\mathrm{CT}$ be the collection of seven axioms on Characterization of Metacategory via Equations.

Then:

$\mathrm{CT} = \mathrm{CT}^*$

where $\mathrm{CT}^*$ consists of the dual statements of those in $\mathrm{CT}$.

## Proof

### Proof for Object Category Theory

The seven axioms are:

 $\ds \operatorname{dom} \operatorname{id}_A = A$ $\qquad$ $\ds \operatorname{cod} \operatorname{id}_A = A$ $\ds f \circ 1_{\operatorname{dom} f} = f$  $\ds 1_{\operatorname{cod} f} \circ f = f$ $\ds \operatorname{dom} \left({g \circ f}\right) = \operatorname{dom} f$  $\ds \operatorname{cod} \left({g \circ f}\right) = \operatorname{cod} g$ $\ds h \circ \left({g \circ f}\right)$ $=$ $\ds \left({h \circ g}\right) \circ f$

Their duals are:

 $\ds \operatorname{cod} \operatorname{id}_A = A$ $\qquad$ $\ds \operatorname{dom} \operatorname{id}_A = A$ $\ds 1_{\operatorname{cod} f} \circ f = f$  $\ds f \circ 1_{\operatorname{dom} f} = f$ $\ds \operatorname{cod} \left({f \circ g}\right) = \operatorname{cod} f$  $\ds \operatorname{dom} \left({f \circ g}\right) = \operatorname{dom} g$ $\ds \left({f \circ g}\right) \circ h$ $=$ $\ds f \circ \left({g \circ h}\right)$

It is seen that only names of the bound variables $f,g$ and $h$ have been changed at some places.

Therefore, we conclude:

$\mathrm{CT}^* = \mathrm{CT}$

$\blacksquare$