Hom-set restrictions and faithful functors $\newcommand{\hom}{\mathrm{hom}}$The CT textbook I'm reading defines faithful functors as follows:

$F: \mathbf A \to \mathbf B$ is called faithful provided that all the hom-set restrictions $$F: \hom_{\mathbf A}(A, A') \to \hom_{\mathbf B}(F A, F A')$$ are injective.

Now the text does not define what a hom-set restriction is exactly, and Google yields poor results. I tried to read along assuming it was a function associating $\hom_{\mathbf A}(A, A')$ in $\mathbf A$ to $\hom_{\mathbf B}(F A, F A')$ in $\mathbf B$, but this conflicts with a subsequent remark about functors being faithful or embeddings.
The book defines embeddings as functors that are injective with respect to morphisms, then states as a remark that:

A functor is an embedding if and only if it is faithful and injective on objects.

If I assume to have guessed what a hom-set restriction is, it doesn't make sense to me that a faithful functor could not be injective on objects, nor can I see how this supposedly stronger condition could imply it being injective on individual morphisms.
So am I wrong somewhere, or what is the proper definition of hom-set restriction?
 A: A functor does several things at once: 


*

*It sends objects in $\def\A{\mathbf A}\A $ to objects in $\def\B{\mathbf B}\B$. If this map $F\colon \def\O{\mathop{\rm Obj}}\O\A \to \O \B$, $A \mapsto FA$ is injective, one calls $F$ injective on objects.

*It sends morphims in $\A$ to morphisms in $\B$. If this map $F \colon \def\M{\mathop{\rm Mor}}\M\A \to \M\B$, $f\colon A \to A'\mapsto Ff \colon FA \to FA'$ is injective, one calls $F$ an embedding.


The mapping of the morphisms can be restricted to the individual hom-sets, recall that 
$$ \M \A = \bigcup_{A, A' \in \O\A} \def\h{\mathop{\rm hom}\nolimits}\h_\A(A,A') $$
giving maps $F \colon \h_\A(A,A') \to \h_\B(FA, FA')$, as restrictions of $F \colon \M\A \to \M\B$. If all these restrictions are injective, one calls $F$ faithful. Note that an embedding is always faithful, but the contrary need not hold: Consider the following example: $\A = (\bullet\, \bullet)$ has two objects and only the two identities as morphisms, $\B = \bullet$ has only one object and one morphism. The obvious (and only) functor $F \colon \A \to \B$ is faithful, as the hom-sets of $\A$ are singletons, but it is not an embedding as the two different identity morphisms in $\A$ are sent to $\B$'s only morphism.
A: The answer depends on definitions of category and functors, but the idea is that functor $F\colon \mathcal C\to\mathcal D$ acts on objects and morphisms:


*

*for every object $c$ in category $\mathcal C$, $Fc$ is object in category $\mathcal D$

*for every two objects $c$ and $c'$ in $\mathcal C$ and every morphism $f\in \operatorname{Hom}_{\mathcal C}(c,c')$, $Ff\in \operatorname{Hom}_{\mathcal D}(Fc,Fc')$


with addition that $F$ preserves identities and composition. One way to formalize this is to define functor as family of functions:


*

*$F\colon \operatorname{Ob}\mathcal C\to\operatorname{Ob}\mathcal D$

*for each pair of objects $c,c'\in\operatorname{Ob}\mathcal C$, function $F_{c,c'}\colon\operatorname{Hom}_{\mathcal C}(c,c')\to \operatorname{Hom}_{\mathcal D}(Fc,Fc')$


When we set up things this way, one usually immediately drops writing indices and writes just $Ff$ instead of $F_{c,c'}f$ for morphism $f\colon c\to c'$. So, when we speak of restriction of $F$ on hom-set $\operatorname{Hom}_{\mathcal C}(c,c')$, we just mean $F_{c,c'}\colon\operatorname{Hom}_{\mathcal C}(c,c')\to \operatorname{Hom}_{\mathcal D}(Fc,Fc')$.
A: I believe I'm reading the same book (The Joy of Cats).  I take a hom-set restriction to be the restriction of a functor to only the morphisms between a particular pair of objects.  
At first, I did not see why the restriction was necessary, and I was tempted to just use these:


*

*A functor is faithful if it is injective on morphisms

*A functor is full if it is surjective on morphisms 


The answers here do a good job explaining why the definitions I have given are not correct.
