On pg 408, proposition 7.6, Rotman states,
If $(B_k)$ is a family of left $R$-modules, then there are natural isomorphisms, for all $n \ge 0$, $$Tor_n^R(A, \bigoplus_k B_k) \cong \bigoplus_k Tor_n^R(A,B_k) $$
But in what "natural" sense?
My thoughts:
We can fix all the $B_ks$
First naturality: we fix all the $B_ks$, then both sdies become functors from R-Mod to Ab. We we prove that this is a natural isomoprhism of functors $Tor_n^R(-, \bigoplus B_k) \cong \bigoplus Tor_n^R(-,B_k)$.
Or we can fix $A$. But then the arguement of the functors should be a collection of $(B_k)_{k \in K}$ of modules. A morphism $(B_k)_{k \in K} \rightarrow (C_j)_{j \in J}$ should be a set of maps $\{ \sigma_{jk}:B_k \rightarrow C_j \}_{k \in K}$. (For each $k$, there is a unique $j$.) This worries me as I am unsure if this forms a category. Fixing $A$, we thus have
Second naturality the isomorphism of functors $Tor_n^R(A, -) \cong \bigoplus Tor_n^R(A,-)$ with arugments $(B_k)_{k \in K}$.
Given a functor $F:A\to B$ between categories with coproducts, consider an indexed family of objects of $A$, say $\{a_i:i\in I\}$, where the indexing set $I$ is fixed. There are maps $a_i\to\bigoplus_i a_i$ which give maps $F(a_i) \to F(\bigoplus_i a_i)$ and hence a natural transformation $\eta : \bigoplus_i F(a_i)\to F(\bigoplus a_i)$. We say $F$ is additive if this map is an isomorphism.
Note that both the left and right hand side can be thought of as functors from diagrams of shape $I$ in $A$ to $B$; the source of $\eta$ is the composition of $A^I\to B^I\to B$ where the first map is the pointwise extension of $F$ and the second is the sum, while the target of $\eta$ is $A^I\to A\to B$, where the first arrow is the sum and the second arrow is $F$.
If you fix one of its arguments $\mathrm{Tor}$ is a usual functor of one variable, and you can prove that it is additive by using the fact tensor products are additive. More generally, you can prove that $\mathrm{Tor}$ commutes with directed (filtered) colimits.