Natural Transformation: Direct Products I have result that tells me $$\displaystyle \varphi : \text{Hom}_R \bigg(A, \prod_{i \in I} B_i \bigg) \to \prod_{i \in I} \text{Hom}_R(A, B_i)$$ is a $Z(R)$-isomorphism. The next result tells me that it is natural (which is the part I'm not understanding):

The isomorphism $\varphi$ is natural: if $(C_j)_{j \in J}$ is a family of left $R$-modules and, for each $i \in I$, there exist $j \in J$ and an $R$-map $\sigma_{ij} : B_i \to C_j$, then there is a commutative diagram
  $\require{AMScd}$
  \begin{CD}
    \text{Hom}_R \bigg(A, \prod_{i \in I} B_i\bigg) @>\varphi>> \prod_{i \in I} \text{Hom}_R(A, B_i)\\
    @V \sigma_* V V@VV \widetilde \sigma V\\
    \text{Hom}_R \bigg( A, \prod_{j \in J} C_j \bigg) @>>\varphi> \prod_{j \in J} \text{Hom}_R(A, C_j)
    \end{CD}
  where $\displaystyle \sigma : \prod_{i \in I} B_i \to \prod_{j \in J} C_j$ is given by $(b_i) \mapsto (\sigma_{ij}b_i)$, and $\sigma_*$ is the induced map, and $\widetilde \sigma : (g_i) \mapsto (\sigma_{ij}g_i)$.

My definition for natural transformation is: 

Let $S, T : \mathcal A \to \mathcal B$ be covariant functors. A natural transformation $\tau : S \to T$ is a one-parameter family of morphisms in $\mathcal B$, $\tau = (\tau_A : SA \to TA)_{A \in \text{Obj(A)}}$ making the following diagram commute for all $f : A \to A'$ in $\mathcal A$:
  $\require{AMScd}$
  \begin{CD}
    SA @>\tau_A>> TA\\
    @V S_f V V @VV T_f V\\
    SA' @>>\tau_{A'}> TA'
    \end{CD}

I'm having trouble figuring out what are $S, T$? Or even $\mathcal A, \mathcal B$?
For instance, am I looking at $\mathcal A$ is a category where the objects are families of left $R$-modules $(B_i)_{i \in I}$? and then $S$ is a functor which takes this category to the category of sets, defined by $$S\big( (B_i)_{i \in I}\big) = \text{Hom}_R\bigg(A, \prod_{i \in I} B_i \bigg)?$$
Or is $\mathcal A$ the category of left $R$-modules where we just look as products as a special case? My concern with this is that maps between products like $\sigma : \prod B_i \to \prod C_j$ are defined in a special way, so where is this taken into account?
To conclude this post, I'm hoping someone can help me see the result in terms of the definition I've been given for natural transformations. (As for showing that the diagram was commutative, that was fairly straight forward.)
Source: "Introduction to Homological Algebra", Joseph Rotman, Page 54, Theorem 2.30.
 A: Ok, I believe that the book try to do the following thing (although avoiding a lot of details).
As a general case for every category $\mathbf C$ you can build a category $\text{Fam}(\mathbf C)$ whose:


*

*objects are families of objects in $\mathbf C$, that is stuff like $(c_i)_{i \in I}$ where each $c_i$ is in $\mathbf C$

*morphisms from $(c_i)_{i \in I}$ to $(d_j)_{j \in J}$ are pairs $\langle f, \sigma \rangle$ where $f \colon J \to I$ and
$\sigma=\langle\sigma_j \colon c_{f(j)} \to d_j\rangle_{j \in J}$

*composition of pair of morphisms
$$(c_i)_{i \in I} \stackrel{\langle f, \sigma\rangle}{\longrightarrow}(d_j)_{j \in J} \stackrel{\langle g,\tau                         
\rangle}{\longrightarrow}(e_l)_{l \in L}$$
is defined to be the morphism
$$(c_i)_{i \in I} \stackrel{\langle h, \psi\rangle}{\longrightarrow} (e_l)_{l \in L}$$
where $h=f \circ g$ and $\psi_{l}=\tau_l \circ \sigma_{g(l)}\colon c_{f(g(l))} \to e_l$ for every $l \in L$.
The identities are the obvious ones.


If $\mathbf C$ is a category with products we have a functor
$$\prod \colon \text{Fam}(\mathbf C) \to \mathbf C$$
that associates to every family $(c_i)_{i \in I}$ (a choice of) its product $\prod_I c_i$.
The arrow part of this functor is not as such straightforward to describe but you can try as an exercise to write it out.
Now back to your problem. As a start we know that for every ring $R$ the category $\mathbf{Mod}_R$ of $R$ modules and $R$-linear maps
has products hence for the observation above you have a functor
$$\prod \colon \mathbf{Fam}(\mathbf {Mod}_R) \to \mathbf{Mod}_R$$
On the other hand for every $R$-module $A$ you also have a functor
$$y_A \colon \textbf{Mod}_R \to \textbf{Mod}_{Z(R)}$$
that sends every module $B$ into the $Z(R)$-module $\text{Hom}_R(A,B)$.
This functor extend naturally to a functor
$$\bar y_A \colon \textbf{Fam}(\textbf{Mod}_R) \to \textbf{Fam}(\textbf{Mod}_{Z(R)})$$
that sends every family $(B_i)_{i \in I}$ of $R$-modules into the family of $Z(R)$-module
$(\text{Hom}_{Z(R)}(A,B_i))_{i \in I}$.
Also the category $\textbf{Mod}_{Z(R)}$ admits its own product functor $\prod$.
From these data you obtain the following functors
$$\mathbf{Fam}(\mathbf{Mod}_R) \stackrel{\prod}{\longrightarrow} \textbf{Mod}_R \stackrel{y_A}{\longrightarrow} \textbf{Mod}_{Z(R)}$$
and
$$\mathbf{Fam}(\mathbf{Mod}_R) \stackrel{\bar y_A}{\longrightarrow} \textbf{Fam}(\textbf{Mod}_{Z(R)})                                  
\stackrel{\prod}{\longrightarrow} \textbf{Mod}_{Z(R)}$$
The first functor associates to each family $(B_i)_{i \in I}$ of $R$-modules the $Z(R)$-module $\text{Hom}_R(A,\prod B_i)$
the second one the module $\prod \text{Hom}_R(A,B_i)$.
The statement of the theorem is saying the family of morphisms $\varphi$ is a natural transformation for these two functors.
