How Should I Prove this Function is an Isomorphism?

Question

I have this question:

Let $L$, $M$ and $N$ be unitary $R$-modules. Let $f: M \to N$ be an $R$-module isomorphism. Prove that the map $f_*: \text{Hom}_R(N,L) \to \text{Hom}_R(M,L)$ is an isomorphism.

This is my approach:

Let $g \in \text{Hom}_R(N,L)$. Define $f_* : \text{Hom}_R(N,L) \to \text{Hom}_R(M,L)$ by $f_*(g(n)) = g(f(x))$ for all $n \in N$ and $x \in M$. Now, consider this diagram:

Notice that it commutes.

I am stuck here as I don’t know how to proceed. I will appreciate any insight you give me on this.

Thanks.

Answer 1

As a hint, you know that $f : M \to N$ is an isomorphism, so it has an inverse $f^{-1} : N \to M$.

Can you show that $f^{-1}_* : \text{Hom}(M,L) \to \text{Hom}(N,L)$ is an inverse to $f_*$?

I'll leave a slightly more descriptive hint below the fold.

Note that $f_*(g) = gf$. So $f^{-1}_* f_* g = g f f^{-1} = g$. Do you see how to finish from here?

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I hope this helps ^_^