We are given $L, M$ and $N$ as unitary $R$-modules, and $f:M\rightarrow N$ as an isomorphism. We need to prove that $f^*:Hom_R(N,L)\rightarrow Hom_R(M,L)$ is an isomorphism.
I started by defining the function f$^*$ this way:
We have f$^*$(g) = gof for any g$\in$Hom$_R$(N,L). Consider the diagram below:
Obviously, the diagram commutes. Since g and f are R-module homomorphisms, then gof is an R-module homomorphism. We have gof $\in$ Hom$_R(M,L)$. Hence, $f^*(g) = gof is well-defined.
My question is, is this okay? Did I do justice to the well-definedness of the function? I have the feeling that I didn’t do a good job here, but I don’t know what to do. Your help will be greatly appreciated.