Difference between faithful and injective on arrows

Question

Could someone explain to me what is the difference between faithful and "injective on arrows" functors? Any help is appreciated.

Answer 1

A faithful functor is only injective with respect to pairs of arrows that had the same domain and codomain to begin with. That is, if $f,g:a\to b$ and $F$ is a faithful functor, $F(f)=F(g)$ implies $f=g$. However, you might have $f:a\to b$ and $g:c\to d$ and $F(a)=F(c)$, $F(b)=F(d)$, and $F(f)=F(g)$, even though $f\neq g$.

For a concrete example, consider the category $C$ with a single object $X$ and only the identity morphism, and the category $D$ with two objects $Y$ and $Z$ and only the identity morphisms. There is a functor $F:D\to C$ that sends $Y$ and $Z$ to $X$ and both identity morphisms to the identity on $X$. Then $F$ is not injective on arrows, since $F(1_Y)=F(1_Z)=1_X$. But $F$ is faithful, since $1_Y$ and $1_Z$ do not have the same domain and codomain.