endofunctor

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

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

• $obj(F):obj(𝓒)\to obj(𝓓)$
• $arr(F):arr(𝓒)\to arr(𝓓)$ Such that:
• $\forall f:arr(𝓒).F(dom(f))=dom(F(f))$
• $\forall f:arr(𝓒).F(cod(f))=cod(F(f))$
• $\forall A,B,C:𝓒;f:A\to B;g:B\to C.F(f\circ g)=F(f)\circ F(g)$ 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.