Definition:Metacategory

Definition
A metacategory $\mathcal C$ consists of:
 * objects $X, Y, Z, \ldots$
 * morphisms or arrows or maps $f, g, h, \ldots$

satisfying the following properties:
 * To each arrow $f$ is associated an object $X = \operatorname{dom} (f)$ called the domain and an object $Y = \operatorname{cod}(f)$ called the codomain . We write write $f:X \to Y$ or $X \stackrel{ f }{\longrightarrow} Y$
 * For every two morphisms $f$ and $g$ such that $\operatorname{cod}(f)=\operatorname{dom}(g)$, the  composition $g \circ f$ or $gf$ is a  morphism with domain  $\operatorname{dom}(f)$ and codomain $\operatorname{cod}(g)$
 * Composition is associative, that is, given $f : X \to Y$, $g: Y \to Z$, $h: Z \to W$, $f\circ(g\circ h)=(f\circ g)\circ h$.
 * For each object $X$ there exists an identity arrow $\operatorname{id}_X$ such that for every  $Y \in \mathcal C_0$ and $f : X\to Y$, $g: Y\to X$ we have  $f\circ\operatorname{id}_X = f$, $\operatorname{id}_X \circ g = g$.

A metacategory is purely axiomatic, and does not use set theory.

For example, the objects are not "elements of the set of objects", because no notion of membership is defined.

An interpretation of the metacategory axioms within set theory is a category