ntral concept which is needed for this purpose is called categorical limit, and can be dualized to yield the notion of a colimit.
It is a natural question to ask, under which conditions two categories can be considered to be "essentially the same", in the sense that theorems about one category can readily be transformed into theorems about the other category. The major tool one employs to describe such a situation is called equivalence of categories. It is given by appropriate functors between two categories. Categorical equivalence has found numerous applications in mathematics.
For example, a (strict) 2-category is a category together with "morphisms between morphisms", i.e. processes which allow us to transform one morphism into another. We can then "compose" these "bimorphisms" both horizontally and vertically, and we require a 2-dimensional "exchange law" to hold, relating the two composition laws. In this context, the standard example is Cat, the 2-category of all (small) categories, and in this example, bimorphisms of morphisms are simply natural transformations of morphisms in the usual sense. Another basic example is to consider a 2-category with a single object—these are essentially monoidal categories. Bicategories are a weaker notion of 2-dimensional categories where the composition of morphisms is not strictly associative, but only associative "up to" an isomorphism.
This process can be extended for all natural numbers n, and these are called n-categories. There is even a notion of ω-category corresponding to the ordinal number ω. For a conversational introduction to these ideas, see Baez (1996).