\begin{definition}[Definition:Metacategory]
A '''metacategory''' is a metagraph subject to extra restrictions.
As such, a '''metacategory''' $\mathbf C$ consists of:
* A collection $\mathbf C_0$ of objects $X, Y, Z, \ldots$
* A collection $\mathbf C_1$ of morphisms $f, g, h, \ldots$ between its objects
The morphisms of $\mathbf C$ are subjected to:
{{begin-axiom}}
{{axiom | n = \text C 1
        | lc= ''Composition''
        | t =
For objects $X, Y, Z$ and morphisms $X \stackrel {f} {\longrightarrow} Y \stackrel {g} {\longrightarrow} Z$ with $\operatorname{cod} f = \operatorname{dom} \, g$, there exists a morphism:
:$g \circ f : X \to Z$
called the composition of $f$ and $g$.
}}
{{axiom | n = \text C 2
        | lc= ''Identity'' 
        | t =
For every object $X$, there is a morphism ${\operatorname{id}_X}: X \to X$, called the identity morphism, such that:
:$f \circ {\operatorname{id}_X} = f$, and ${\operatorname{id}_X} \circ g = g$
for any object $Y$ and morphisms $f : X \to Y$ and $g : Y \to X$
}}
{{axiom | n = \text C 3
        | lc= ''Associativity''
        | t =
For any three morphisms $f, g, h$:
:$f \circ \paren {g \circ h} = \paren {f \circ g} \circ h$
whenever these compositions are defined (according to $(\text C 1)$).
}}
{{end-axiom}}
To describe a '''metacategory''' it is necessary to specify:
:* The collection $\mathbf C_0$ of objects
:* The collection $\mathbf C_1$ of morphisms
:* For each object $X$, an identity morphism ${\operatorname{id}_X}: X \to X$
:* For every appropriate pair of morphisms $f, g$, the composite morphism $g \circ f$
However, the last two are often taken to be implicit when the objects and morphisms are familiar.
Of course, after defining these, it is still to be shown that $(\text C 1)$ up to $(\text C 3)$ are satisfied.
A '''metacategory''' is purely axiomatic, and does not use set theory.
For example, the objects are not "elements of the set of objects", because these axioms are (without further interpretation) unfounded in set theory.
For some purposes, it is convenient to have a different description of a '''metacategory'''. Two such descriptions are found on:
* Characterization of Metacategory via Equations
* Morphisms-Only Metacategory
\end{definition}