Hopfian modules and equivalence of categories of modules For a ring with unity (not necessarily commutative) $R$, let $R$-$Mod$ denote the category of left $R$-modules. 
Let $R,S$ be two rings with unity and $T: R$-Mod $\to S$-Mod be an equivalence of categories  ($T$ be co-variant). Is it true that $M$ is a Hopfian $R$-module if and only if $T(M)$ is an Hopfian $S$-module ?  
In general, if $R$-Mod and $S$-Mod are equivalent as categories, then do we have a one-to-one correspondence between Hopfian $R$-modules and Hopfian $S$-modules ?  

Yes, this is trivial: a module $M$ is Hopfian iff every epimorphism $M\to M$ is an isomorphism, which is preserved by any equivalence of categories.
