# Axiom:Axioms for Morphisms-Only Category Theory

Let $\mathcal{MOCT}$ be the language of (morphisms-only) category theory.
 $(MOCT0)$ $:$ $\displaystyle \forall x,y,z,z':$ $\displaystyle \left({R_\circ \left({x, y, z}\right) \land R_\circ \left({x, y, z'}\right)}\right) \implies z = z'$ $\circ$ 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 $\circ$