Functor

A functor is a function that maps objects and arrows between two categories preserving structure.

Given $\mathcal{C}$, $\mathcal{D}$ : Cat,
A functor $F:\mathcal{C}\to \mathcal{D}$ is a pair of maps:

• $obj(F):obj(\mathcal{C})\to obj(\mathcal{D})$
• $arr(F):arr(\mathcal{C})\to arr(\mathcal{D})$
  Such that:
• $\forall f:arr(\mathcal{C}).F(dom(f))=dom(F(f))$
• $\forall f:arr(\mathcal{C}).F(cod(f))=cod(F(f))$
• $\forall A,B,C:\mathcal{C};f:A\to B;g:B\to C.F(f\circ g)=F(f)\circ F(g)$
  In words, $f$ preserves domain, codomain and composition. Preserving identity follows from this.

An endofunctor is just a functor where the domain equals the codomian.

Remarks

Not all images of a functor are a category. See Image of a Functor.