TITLE: Homomorphisms vs Functors
QUESTION [13 upvotes]: What really is the difference between a Homomorphism between categories and a Functor? I understand that they are both structure preserving mapping of the objects and arrows between the categories or am i missing something ?

REPLY [7 votes]: If I understand your question and definitions correctly: no, a homomorphism of categories and a functor are not the same thing. 
Your definition of "homomorphism of categories" seems to be: given categories $M$ and $N$, a homomorphism $h:M\to N$ is 

a mapping from $\mathsf{ob}(M)$ to $\mathsf{ob}(N)$, together with
for every $x,y\in\mathsf{ob}(M)$, a mapping from $\mathrm{Hom}_M(x,y)$ to $\mathrm{Hom}_N(h(x),h(y))$ such that $$h(a\circ b)=h(a)\circ h(b)$$ for all $a,b\in \mathrm{Hom}_M(x,y)$.

A functor is then a homomorphism of categories that satisfies the additional requirement that

$h(\mathrm{id}_x)=\mathrm{id}_{h(x)}$ for all $x\in\mathsf{ob}(M)$.

Thus, being a functor is a strictly stronger property than being a homomorphism of categories. For example, let $M$ be the category with 

$\mathsf{ob}(M)=\{x,y\}$
$\mathrm{Hom}_M(x,x)=\{\mathrm{id}_x\},\quad \mathrm{Hom}_M(x,y)=\{a\}, \quad \mathrm{Hom}_M(y,x)=\varnothing,\quad \mathrm{Hom}_M(y,y)=\{\mathrm{id}_y\}$

and let $N$ be the category with

$\mathsf{ob}(N)=\{z\}$
$\mathrm{Hom}_N(z,z)=\{\mathrm{id}_z,c\}$ where $c\circ c=c$.

Then the homomorphism of categories $h:M\to N$ defined by

$h(x)=z,\quad h(y)=z$
$h(\mathrm{id}_x)=c,\quad h(a)=c,\quad h(\mathrm{id}_y)=c$

is not a functor.