I've encountered this problem on my Non commutative algebra handouts wich says:
given $R,S$ rings and $f:R\to S\:$ a ring homomorphism, define a canonical functor $F:\textbf{Mod-S}\to \textbf{Mod-R}.$ Where $\textbf{Mod-S}$ is the category of right $S-$modules and similarly $\textbf{Mod-R}$ is the category of right $R-$modules.
Once you've done this, prove that the functor $F$ is faithful.
Now, my problem is that the definition of $F$ doesn't sound natural at all to me. It would be better in my opinion to define $F:\textbf{Mod-R}\to \textbf{Mod-S}.$ Am i wrong? and in the case, how to define $F$?
Thank you for your replies.