Injection, full, monic, faithful, conservative in category theory

The concepts named in the subject line are related, even interdefinable, and provide examples of each other. $\Newextarrow{\xLeftarrow}{2,2}{0x21D0}$
The purpose of this post is to show some of those interrelations.

\[\begin{array}{cccl} \setX & \xrightarrow{\textstyle \mkern1em \functionf \mkern1em} & \setY && \text{a function in $\Set$} \\ (\eltx = \eltxp) & \xLeftarrow{} & (\eltx\functionf = \eltxp\functionf) && \text{$\functionf$ is $\textit{injective}$ iff, $\forall \eltx, \eltxp\in\setX$, this logical implication holds } \\ \hom \eltx \setX \eltxp & \xrightarrow {\textstyle \hom \eltx \functionf \eltxp} & \hom {\eltx\functionf} \setY {\eltxp\functionf} && \text{the arrow in $\calV$ between the hom-objects} \\ \end{array}\]

If we consider $\setX \xrightarrow{\textstyle \mkern1em \functionf \mkern1em} \setY$ as, not (a function in $\Set$), but as (a $\calV$-functor between discrete categories enriched in either $\calV=\bftwo$ or $\calV=\Set$),
then this condition amounts to saying that the arrow in the third line, in $\calV$, is invertible, i.e.,
that (the $\calV$-functor $\functionf$ between discrete $\calV$-categories) is not merely faithful, as it automatically is in those cases, but also full, i.e.,
that $\functionf$ is a full and faithful, or fully faithful $\calV$-functor.