Endomorphisms of an object in a category. Arguments order in multiplication agrees with function.comp, not with category.comp.

Multiplication of endomorphisms agrees with function.comp, not category_struct.comp.

Endomorphisms of an object form a monoid

In a groupoid, endomorphisms form a group

f.map as a monoid hom between endomorphism monoids.

f.map_iso as a group hom between automorphism groups.