Category Axioms are Self-Dual

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 for Object Category Theory
The seven axioms are:

Their duals are:

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

Therefore, we conclude:


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