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 $\tuple {g, f}$ with $\operatorname {cod} f = \operatorname {dom} g$; write $\map {\mathbf C_2} {g, f}$ to express that $\tuple {g, f}$ is a member of $\mathbf C_2$.

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

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

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

Axioms $(MOCT1)$ up to $(MOCT3)$ combine to ensure that $h \circ \paren {g \circ f}$ is defined $\paren {h \circ g} \circ f$ 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