Theory of functors

The category of locally small categories is a concrete category. As a consequence of this, every category has an underlying set and every functor has an underlying function. As functions are very well understood objects, we can understand functors by reference to their underlying functions. By this treatment, functors are simply the homomorphism functions of categories.

Parts referred to by functors:
The homomorphisms for basic algebraic structures like monoids are well known. These show that the kernel of any homomorphism is a congruence and the image is a subalgebra. We will now present the analagous result for categories.

• The image of a functor is a categorical system of a subcategory.
• The kernel of a functor is a categorical congruence.

The image is embedded in a subcategory by closure, as is the case for any categorical system. If a functor is object injective, then its image is necessarily a subcategory. These two homomorphism theorems for categories relate functors to the lattices of subalgebras $Sub(C)$ and congruences $Con(C)$ of a category.

Parts of functors:
A functor is a type of function, which means it is also an algebraic structure in its own right with its own subobjects and quotients. Subobjects are restrictions, and quotients are input/output relationships where in some partition of the input determines a partition of the output.

Every functor can be restricted to an object function and a morphism function. A functor is then simply a coproduct of its object and morphism functions. On the most basic level, the subobjects of a functor are individual ordered pairs of categorical elements from the two categories. Another case when you might get a subobject of a functor is by restriction to a subcategory of the domain.

Quotients are generally more interesting because they can tell you more about the functioning of a functor. There are some quotient relationships inherent to any functor. Recall the following hierarchy of properties of a morphism. Then if we have the morphism function part of a functor we have that the input object of the input of the functor determines the input object of the output of the functor. Likewise, the output object of the input determines the output object of the output. In both cases, the quotient function is the object function. By combination of these, we can also get that input output pairs of morphism determines input output pairs of their output.

Limits and colimits of categories:
The concrete category $Cat$ of locally small categories $Cat$ has all small limits/colimits. In particular, we can form products and coproducts of categories. The coproduct of two categories has the objects in the two categories in separate connected components. The product has as morphisms ordered pairs of morphisms of the two categories with composition between them piecewise.

Universal algebraic properties of categories:
In conclusion, categories have all the trappings of any algebraic structure studied in universal algebra. They have underlying sets, homomorphisms, subalgebras, congruences, free objects, products, coproducts, varieties and pseudovarieties, and so on. The elementary theory of categories is provided by constructions of universal algebra applicable to any algebraic structure.