Suppose we have an isomorphism of projective curves $f : C \to C'$. Let $M_C(\mu)$ denote the moduli space of slope stable vector bundles on $C$ with slope $\mu$. Do we get an induced isomorphism $F : M_C(\mu) \to M_{C'}(f(\mu))$?
If $f_*$ is the map induced by $f$ on the level of, say, coherent sheaves (or even derived categories), then I think I can show that if $E$ is a vector bundle on $C$ then $f_*(E)$ is a vector bundle on $C'$ in my case.
However, I'm not sure what to do about slope stability. I tried the following: let $E \in M_C(\mu)$ and consider a sub-vector bundle $S \subset f_*(E)$ with slope $\mu(S) \geq \mu(f_*(E))$. Now $\mathrm{Hom}(S, f_*(E)) \cong \mathrm{Hom}(f^*(S), E)$ by adjunction. Then I said something like "slope behaves well under pull-back", which would mean that $\mu(f^*(S)) \geq \mu(E)$ and by slope stability of $E$, the Hom would vanish giving slope stability of $f_*(E)$, but I think this argument is way too simplistic (i.e. I don't think the slope being preserved under pullback thing is true, because there is another question on MathOverflow asking about the stability of pushforwards of line bundles on curves, and the answer seems non-trivial).
I'd be grateful for any help, or pointers to a reference if there is a simple way of proving this statement. Thanks.