In every book I've looked, the "naturality" of the long exact sequence in homology simply says that every arrow between short exact sequences translates into an arrow between the long exact sequences they induce.
Formally speaking, doesn't this mean the long exact sequence is functorial in the input short exact sequence? Did the term naturality just stick historically, or is there some actual natural transformation whose components are the assignments of long exact sequences to short exact sequences?
Best Answer
Let's setup the framework. Let $\mathsf{Ab}$ be the category of abelian groups, and let $\mathsf{SES}$ be the category of short exact sequences, that is diagrams of the form $0 \to A \xrightarrow{f} B \xrightarrow{g} C \to 0$ (with the usual conditions about images and kernels) and morphisms the obvious thing. Then define three functors, $$\begin{align} (H_n^a, H_n^b, H_n^c) & : \mathsf{SES} \to \mathsf{Ab}^3 \\ (A,B,C,f,g) & \mapsto (H_n(A), H_n(B), H_n(C)) \end{align}$$
The functoriality simply means that if you have three exact sequences and two chain maps $X \xrightarrow{\phi} X' \xrightarrow{\psi} X''$, then $H_n^x(\psi \circ \phi) = H_n^x(\psi) \circ H_n^x(\phi)$ for $x = a,b,c,$.
On the other hand the long exact sequence construction is a combination of three families of natural transformations, namely $f_* : H_n^a \to H_n^b$, $g_* : H_n^b \to H_n^c$ (both defined in the obvious way), and the connecting morphism $\partial_* : H_n^c \to H_{n-1}^a$. When one says that the long exact sequence is natural, one is really saying that these three families are all natural transformations.
Of course, you can set things up differently and construct a functor $\operatorname{LES} : \mathsf{SES} \to \mathsf{Ch}$ from the category of short exact sequences to the category of chain complexes, bundling everything together. But when people say that the long exact sequence construction is natural, they are implicitly thinking about all the $H_n$ separately.
This is in fact somewhat of a general construction. If you have two functors $F,G : \mathsf{C} \to \mathsf{D}$ and a natural transformation $\phi : F \to G$, then you can build up a new functor $\mathsf{C} \to \mathsf{Ar}(\mathsf{D})$ given by: $$c \in \mathsf{C} \mapsto (F(c) \xrightarrow{\phi_c} G(c)) \in \mathsf{Ar}(\mathsf{D}),$$ and vice-versa (a functor $\mathsf{C} \to \mathsf{Ar}(\mathsf{D})$ is the same thing as two functors and a natural transformation).
It's hard to say which one is more "natural", but in general using this principle you can interpret any naturality result as a functoriality result, if you want – at the cost of using a more complicated target category.