Axiom:Axioms for Morphisms-Only Category Theory

Axiom

Then (morphisms-only) category theory is the mathematical theory arising from the following axioms:

$(MOCT0)$	$:$	$\displaystyle \forall x,y,z,z':$	$\displaystyle \left({R_\circ \left({x, y, z}\right) \land R_\circ \left({x, y, z'}\right)}\right) \implies z = z' $	$\circ$ is a partial mapping in two variables
$(MOCT1)$	$:$	$\displaystyle \forall x,y:$	$\displaystyle \operatorname{dom} x = \operatorname{cod} y \iff \exists z: R_\circ \left({x, y, z}\right) $	domain of composition $\circ$
$(MOCT2)$	$:$	$\displaystyle \forall x,y,z:$	$\displaystyle R_\circ \left({x, y, z}\right) \implies \left({\operatorname{dom} z = \operatorname{dom} y \land \operatorname{cod} z = \operatorname{cod} x}\right) $	Domain and codomain of a composite $z = x \circ y$
$(MOCT3)$	$:$	$\displaystyle \forall x,y,z,a,b:$	$\displaystyle R_\circ \left({x, y, a}\right) \land R_\circ \left({y, z, b}\right) \implies \left({\exists w: R_\circ \left({x, b, w}\right) \land R_\circ \left({a, z, w}\right)}\right) $	$\circ$ is associative
$(MOCT4)$	$:$	$\displaystyle \forall x:$	$\displaystyle R_\circ \left({x, \operatorname{dom} x, x}\right) \land R_\circ \left({\operatorname{cod} x, x, x}\right) $	Left-identity and right-identity for $\circ$

\((MOCT0)\)	$:$	\(\displaystyle \forall x,y,z,z':\)	\(\displaystyle \left({R_\circ \left({x, y, z}\right) \land R_\circ \left({x, y, z'}\right)}\right) \implies z = z' \)	$\circ$ is a partial mapping in two variables
\((MOCT1)\)	$:$	\(\displaystyle \forall x,y:\)	\(\displaystyle \operatorname{dom} x = \operatorname{cod} y \iff \exists z: R_\circ \left({x, y, z}\right) \)	domain of composition $\circ$
\((MOCT2)\)	$:$	\(\displaystyle \forall x,y,z:\)	\(\displaystyle R_\circ \left({x, y, z}\right) \implies \left({\operatorname{dom} z = \operatorname{dom} y \land \operatorname{cod} z = \operatorname{cod} x}\right) \)	Domain and codomain of a composite $z = x \circ y$
\((MOCT3)\)	$:$	\(\displaystyle \forall x,y,z,a,b:\)	\(\displaystyle R_\circ \left({x, y, a}\right) \land R_\circ \left({y, z, b}\right) \implies \left({\exists w: R_\circ \left({x, b, w}\right) \land R_\circ \left({a, z, w}\right)}\right) \)	$\circ$ is associative
\((MOCT4)\)	$:$	\(\displaystyle \forall x:\)	\(\displaystyle R_\circ \left({x, \operatorname{dom} x, x}\right) \land R_\circ \left({\operatorname{cod} x, x, x}\right) \)	Left-identity and right-identity for $\circ$