@LastIronStar If I am reading this correctly, for faithful functor we do not require $F_{X,Y}\colon\mathrm{Hom}_{\mathcal C}(X,Y)\rightarrow\mathrm{Hom}_{\mathcal D}(F(X),F(Y))$ to be surjective.
For example consider $\mathbf{Top}\to\mathbf{Set}$.
Every morphism in $\mathbf{Top}$ (=continuous map) is a morphism in $\mathbf{Set}$.
But if you have two topological spaces, there are many functions $X\to Y$ which are not continuous.
BTW Roman gives also this warning:
> We should note that the term embedding, as applied to functors, is defined differently by different authors. Some authors define an embedding simply as a full and faithful functor. Other authors define an embedding to be a faithful functor whose object part is injective. We have adopted the strongest definition,
The paragraph in Roman following the definitions of full/faithful/embedding seems to be mainly about the fact that faithful functor is not necessarily injective on objects. (So what he discussed there is without the injectivity condition which you added.)
Let's hope that somebody who knows more about this area will notice your question about text recommendation. The best advice I am able to offer is to look at posts on the main site tagged book-recommendation+category-theory.
Since I have to leave, I'll leave this discussion for others. But I wanted at least add a quick response to your inquiry. (Maybe somebody gives you a bit more detailed/insightful answer.)
It's not the same thing, but there are some books about category which are directed at computer scientist and are written with that audience in mind. But I am not sure whether this close to what you meant by text geared towards discrete mathematics.
There is a book Pierce B.C. Basic category theory for computer scientists. I did not read it - but the title suggests it might be suitable for computer scientists.
The book Asperti A., Longo G. Categories, types, and structures (An Introduction to Category Theory for the working computer scientist) was mentioned a few times in book recommendations on category theory.
So was Awodey: Category Theory.
Based on a brief glance, both Awodey and Asperti-Longo seems rather advanced to me. (Although maybe introductory chapters can be read by beginners?)
But it is possible that I am answering something different from what you want - the key difference being emphasis on computer science vs. emphasis on discrete math.
Let us hope somebody more experienced in this topic will notice your question and respond here.
@MartinSleziak I looked through some of the resources you've gleaned and shared. I think these draw on examples specific to the domain of computer science like programming theory, etc,.
Category Theory
Category Theory