# Axiom:Axioms for Morphisms-Only Category Theory

Let $\mathcal {MOCT}$ be the language of (morphisms-only) category theory.
 $(\text {MOCT} 0)$ $:$ $\ds \forall x, y, z, z':$ $\ds \paren {\map {R_\circ} {x, y, z} \land \map {R_\circ} {x, y, z'} } \implies z = z'$ $\circ$ is a partial mapping in two variables $(\text {MOCT} 1)$ $:$ $\ds \forall x, y:$ $\ds \Dom x = \Cdm y \iff \exists z: \map {R_\circ} {x, y, z}$ domain of composition $\circ$ $(\text {MOCT} 2)$ $:$ $\ds \forall x, y, z:$ $\ds \map {R_\circ} {x, y, z} \implies \paren {\Dom z = \Dom y \land \Cdm z = \Cdm x}$ Domain and codomain of a composite $z = x \circ y$ $(\text {MOCT} 3)$ $:$ $\ds \forall x, y, z, a, b:$ $\ds \map {R_\circ} {x, y, a} \land \map {R_\circ} {y, z, b} \implies \paren {\exists w: \map {R_\circ} {x, b, w} \land \map {R_\circ} {a, z, w} }$ $\circ$ is associative $(\text {MOCT} 4)$ $:$ $\ds \forall x:$ $\ds \map {R_\circ} {x, \Dom x, x} \land \map {R_\circ} {\Cdm x, x, x}$ Left identity and right identity for $\circ$