A small $2$ -category, $\mathcal{C}_2$ , is the first of higher order n-categories constructed as follows.

1. Define $\mathcal{C}at$ as the category of small categories and functors
2. Define a class of objects $A, B,...$ in $\mathcal{C}at$ called `$0$ - cells'
3. For all `$0$ -cells' $A$ , $B$ , consider a set denoted as ``$\mathcal{C}_2 (A,B)$ '' that is defined as $\hom_{\mathcal{C}_2}(A,B)$ , with the elements of the latter set being the functors between the $0$ -cells $A$ and $B$ ; the latter is then organized as a small category whose $2$ -`morphisms', or `$1$ -cells' are defined by the natural transformations $\eta: F \to G$ for any two morphisms of $\mathcal{C}at$ , (with $F$ and $G$ being functors between the `$0$ -cells' $A$ and $B$ , that is, $F,G: A \to B$ ); as the `$2$ -cells' can be considered as `$2$ -morphisms' between $1$ -morphisms, they are also written as: $\eta : F \Rightarrow G$ , and are depicted as labelled faces in the plane determined by their domains and codomains
4. The $2$ -categorical composition of $2$ -morphisms is denoted as ``$\bullet$ '' and is called the vertical composition
5. A horizontal composition, ``$\circ$ '', is also defined for all triples of $0$ -cells, $A$ , $B$ and $C$ in $\mathcal{C}at$ as the functor $$\circ: \mathcal{C}_2(B,C) \times \mathcal{C}_2(A,B) = \mathcal{C}_2(A,C),$$ which is associative
6. The identities under horizontal composition are the identities of the $2$ -cells of $1_X$ for any $X$ in $\mathcal{C}at$
7. For any object $A$ in $\mathcal{C}at$ there is a functor from the one-object/one-arrow category ${1}$ (terminal object) to $\mathcal{C}_2(A,A)$ .