Confusion regarding definition of adjoint functor - Hilton and Stammbach

While defining Adjoint functors in their book A Course in Homological Algebra, Hilton and Stammbach said the follwing:

Let $F:\mathfrak{S}\rightarrow \mathfrak{M}_{\Lambda}$ be the free functor which associates with every set the free $\Lambda$ module on that set as a basis; and let $G:\mathfrak{M}_{\Lambda}\rightarrow\mathfrak{S}$ be the underlying functor which associates with every module its underlying set. We now define a transformation, natural in both $S$ and $A$, $$\eta=\eta_{SA}:\mathfrak{M}_{\Lambda}(FS,A)\rightarrow \mathfrak{S}(S,GA)$$

I did not understand what does he mean by natural in both $S$ and $A$..

I know that a natural transformation is between two functors $F,G: \mathcal{C}\rightarrow \mathcal{D}$..

What are those two functors here...

Fix $A$ a $\Lambda$ module. Then we have functor $$\mathfrak{M}_{\Lambda}(F-,A):S\mapsto \mathfrak{M}_{\Lambda}(FS,A)$$ from sets to module homomorphisms

Fix $S$ a set. Then we have a functor $$\mathfrak{S}(S,G-):A\mapsto \mathfrak{S}(S,GA)$$ from modules to set maps..

To define natural transformations, we need to have two functors from same categor landing in same category... Here one functor is from category of sets and other is from category of modules... What natural equivalence are we considering here?

I am very much confused..

There seems to be a misunderstanding here. Your $\mathfrak M_Λ(F-, A)$ is not a functor from sets to "module homomorphisms", it is a contravariant functor from sets to sets, which assigns to every set $S$ a set of certain module homomorphisms. In other words, it is a functor from $\mathfrak S^\mathrm{op}$ to $\mathfrak S$, as is $\mathfrak S(-, GA)$. That $η_{SA}$ is natural in $S$ means that for a fixed $A$, $η_{-,A}$ is a natural transformation between these two functors, and similarly for naturality in $A$.
Another way to say this is to note that $(S, A) ↦ \mathfrak M_Λ(FS, A)$ is a functor $\mathfrak S^\mathrm{op} × \mathfrak M_Λ → \mathfrak S$, and so is $(S, A) ↦ \mathfrak S(S, GA)$, and you can check that $η$ is natural in both $S$ and $A$ exactly when it is a natural transformation between them.
• I am still getting confused... Why do we have $\mathfrak{S}^{\rm{op}}$ there and not just $\mathfrak{S}$ – kk lm Feb 3 '16 at 19:04