If I have a functor $F:_RMod\rightarrow _SMod$ that is between the category of $R$-modules to the category of $S$-modules. Can I show that it must be the case that $F(0)=0$.
I know that $_RMod$ and $_SMod$ are both preaddative categories. So can I use this to show that every functor must have this property?
If we have that the map $Id_0:0\rightarrow 0$ is the identity map on $0$ then we know that $F(Id_0)=Id_{F(0)}$ but this doesn't seem like enough.
Thanks for any help