Editing Functor (section)

== Properties ==
Two important consequences of the functor [[axiom]]s are:
* ''F'' transforms each [[commutative diagram]] in ''C'' into a commutative diagram in ''D'';
* if ''f'' is an [[isomorphism]] in ''C'', then ''F''(''f'') is an isomorphism in ''D''.

One can compose functors, i.e. if ''F'' is a functor from ''A'' to ''B'' and ''G'' is a functor from ''B'' to ''C'' then one can form the composite functor {{nowrap|''G'' ∘ ''F''}} from ''A'' to ''C''. Composition of functors is associative where defined. Identity of composition of functors is the identity functor. This shows that functors can be considered as morphisms in categories of categories, for example in the [[category of small categories]].

A small category with a single object is the same thing as a [[monoid]]: the morphisms of a one-object category can be thought of as elements of the monoid, and composition in the category is thought of as the monoid operation. Functors between one-object categories correspond to monoid [[homomorphism]]s. So in a sense, functors between arbitrary categories are a kind of generalization of monoid homomorphisms to categories with more than one object.