Definition:Morphisms-Only Metacategory

Definition
A morphisms-only metacategory is a metamodel for the language of category theory subject to the following axioms:

Explanation
A morphisms-only metacategory can thus be described as follows.

Let $\mathbf C_1$ be a collection of objects called morphisms.

Let $\mathbf C_2$ be the collection of pairs of morphisms $\left({g, f}\right)$ with $\operatorname{cod} f = \operatorname{dom} g$; write $\mathbf C_2 \left({g, f}\right)$ to express that $\left({g, f}\right)$ is a member of $\mathbf C_2$.

By $(MOCT1)$, we see that $\mathbf C_2 \left({g, f}\right)$ thus is an abbreviation of the statement $\exists h: R_\circ \left({g, f, h}\right)$.

Let $\circ$ be an operation symbol which must assign to every pair of morphisms $\left({g, f}\right)$ in $\mathbf C_2$ a morphism $g \circ f$, called the composition of $g$ with $f$.

We see that $g \circ f$ satisfies $R_\circ \left({g, f, g \circ f}\right)$; by axiom $(MOCT0)$, it is unique.

Axioms $(MOCT1)$ up to $(MOCT3)$ combine to ensure that $h \circ \left({g \circ f}\right)$ is defined iff $\left({h \circ g}\right) \circ f$ is, and that they are equal when this is the case.

Finally, axiom $(MOCT4)$ entails the existence and uniqueness of left- and right-identities for $\circ$.

Also see

 * Morphisms-Only Metacategory Induces Metacategory
 * Metacategory Induces Morphisms-Only Metacategory