Faithful, full, fully faithful functors.

Question

In all textbooks I read, a functor $F\colon\mathcal{A}\to\mathcal{B}$ is said to be faithful if, for every two objects $A,A'$ of $\mathcal{A}$, the function $F_{AA'}\colon\operatorname{Hom}_{\mathcal{A}}(A,A') \to \operatorname{Hom}_{\mathcal{B}}(FA,FA')$, defined by $F_{AA'}(f)=F(f)$ is injective. And similarly for full functors. I can't understand why that definition is not simplified by the following one: a functor $F\colon\mathcal{A}\to\mathcal{B}$ is faithfull if $F_1$ is injective, meaning by $F_1$ the function $F_1\colon\mathcal{A}_1\to\mathcal{B}_1$, and by $\mathcal{A}_1$ and $\mathcal{B}_1$ the classes of morphisms of $\mathcal{A}$ and $\mathcal{B}$ respectively. Is this second definition wrong?

Answer 1

Yes, it is wrong. Your suggestion is actually more restrictive than the usual definition, because in the usual definition we allow maps between different objects, for example $f:A\to B$ and $f':A'\to B'$, to be mapped to the same $g:X\to Y$ if $F(A)=X=F(A')$ and $F(B)=Y=F(B')$, which your suggestion doesn't allow. In fact your suggestion would imply that any full functor must be injective on objects (because different identities could not be mapped to the same identity); so for example with your definition the forgetful functor $\mathbf{Top}\to \mathbf{Set}$ would not be faithful. In fact, almost none of the usual forgetful functors would be faithful!