nLab
function

Idea

A function is a homomorphism of sets. Functions are the morphisms in the category Set.

A function is sometimes called a total function to distinguish it from a partial function.

See set theory for more details.

As morphisms of discrete categories

If we regard sets as discrete categories, then a function is a functor between sets. The functoriality structure becomes the property that a function preserves equality:

(1)$$x=y\Rightarrow f(x)=f(y).$$x = y \Rightarrow f(x) = f(y) .

For classes

See the MathOverflow: what-are-maps-between-proper-classes