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Definition 0.1.
A small $2$category, $\mathcal{C}_{2}$, is the first of higherorder ncategories constructed as follows.
1. Define $\mathcal{C}at$ as the category of small categories and functors
2. 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 oneobject/onearrow category 1 (terminal object) to $\mathcal{C}_{2}(A,A)$.
0.1 Examples of 2categories
1. The $2$category $\mathcal{C}at$ of small categories, functors, and natural transformations;
2. The $2$category $\mathcal{C}at(\mathcal{E})$ of internal categories in any category $\mathcal{E}$ with finite limits, together with the internal functors and the internal natural transformations between such internal functors;
3. When $\mathcal{E}=\mathcal{S}et$, this yields again the category $\mathcal{C}at$, but if $\mathcal{E}=\mathcal{C}at$, then one obtains the 2category of small double categories;
4. When $\mathcal{E}=\textbf{Group}$, one obtains the $2$category of crossed modules.
0.2 Remarks

In a manner similar to the (alternative) definition of small categories, one can describe $2$categories in terms of $2$arrows. Thus, let us consider a set with two defined operations $\otimes$, $\circ$, and also with units such that each operation endows the set with the structure of a (strict) category. Moreover, one needs to assume that all $\otimes$units are also $\circ$units, and that an associativity relation holds for the two products:
$(S\otimes T)\circ(S\otimes T)=(S\circ S)\otimes(T\circ T);$ 
A $2$category is an example of a supercategory with just two composition laws, and it is therefore an $\S_{1}$supercategory, because the $\S_{0}$ supercategory is defined as a standard ‘$1$’category subject only to the ETAC axioms.
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