Definition:Functor

Definition
A functor is a morphism of categories.

Co- and Contravariance
Both covariant and contravariant functors are paramount in all of contemporary mathematics.

The intention behind defining a functor is to formalise and abstract the intuitive notion of "preserving structure".

Functors thus can be understood as a generalisation of the concept of homomorphism in all its instances.

This explains why one would be led to contemplate covariant functors.

However, certain "natural" operations like transposing a matrix do not preserve the structure as rigidly as a homomorphism (we do have Transpose of Matrix Product, however).

Because of the abundant nature of this type of operations, the concept of a contravariant functor was invented to capture their behaviour as well.