I am confused about what exactly I have to check or how to check when a problem asks that there is a natural isomorphism.
My confusion arises in the following problem from Lang's Algebra (Chapter XVI Ex 4):
Let $\phi:A\rightarrow B$ be a commutative ring homomorphism. Let $E$ be an $A$-module and $F$ a $B$-module. Let $F_A$ be the $A$-module obtained from $F$ via the operation of $A$ on $F$ through $\phi$, that is for $y\in F_A$ and $A\in A$ this operation is given by $(a,y)\rightarrow \phi(a)y$ Show there is a natural isomorphism $\text{Hom}_B(B\otimes_A E,F)\cong \text{Hom}_A(E,F_A)$
I am confused firstly about when an isomorphism means in this context because the left hand side is a $B$ module and the right hand side an $A$ module. Secondly I know the definition of natural transformation has to do with functors, but I do not see how this applies or what I have to check to show that an isomorphism is natural in this context.
Thank you all.