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If it wasn’t clear the way it was, then I suggest to expand on it while sticking to the notation that the paragraph starts with. I have made it show this (here):
$\array{ \mathcal{C}(x,y) &\xhookrightarrow{\;\; F_{x,y} \;\;}& \mathcal{D}(F(x), F(y)) \\ (x \overset{\phi}{\to} y) &\mapsto& \big( F(x)\overset{F(\phi)}{\to} F(y) \big) \,. }$Also I have removed the side-remark you added that faithful functors are called “embeddings”, because that terminology is fraught with issues. Instead I have added under “Related concepts” a pointer to embedding of categories, where this is discussed in more detail.
I’m not sure that 0-truncated morphisms are the right thing to list under generalizations of the notion of faithful functor.
It seems a reasonable characterization for functors between infinity groupoids, since $X \to Y$ being 0-truncated means $X(x,x) = \Omega_x X \to \Omega_y Y = Y(y,y)$ is a monomorphism.
But it doesn’t seem reasonable for functors between infinity categories. For example, if $C$ is a 1-category with no nontrivial isomorphisms, then when viewing $C$ as an infinity category, $C \to *$ is a 0-truncated morphism of $\infty Cat$ (since the nerve of $C$ is a complete segal space whose levels are all sets). But $C \to *$ need not be a faithful functor in the sence of 1-categories.
I would expect “locally monic” would be the right generalization to infinity categories, and dually “locally an effective epimorphism” would be the right generalization of “full”.
Maybe just say that 0-truncated morphisms in an $(\infty,1)$-category are a generalization of faithful functors between groupoids?
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