I just noticed something funky. Let $X$ denote an $I$-indexed family of sets. There is a projection $$\pi_X: \bigsqcup_{i:I} X_i \rightarrow I.$$ It isn't necessarily surjective, of course, because one or more of the $X_i$ may be empty. Anyway, I noticed that the set $\prod_{i:I} X_i$ can be identified with the set of sections of $\pi_X$; that is, with the set of all $m : I \rightarrow \bigsqcup_{i:I} X_i$ satisfying $\pi_X \circ m = \mathrm{id}_I$.
This is intuitively obvious and probably straightforward to prove. Nonetheless, I find the existence of this kind of a connection between products and coproducts to be pretty surprising.
Question. I'd like some "intellectual context" for this observations.
For example, I'm interested in: related facts, natural generalizations, important consequences, interesting applications, etc.
Basically, I'd like someone to do here what I tried to do for the OP here.
Let $C$ be a category containing a final objects, products and coproducts either finite or arbitrary. Let $1$ denote the final object. For $A$ an object of $C$, the space $Hom(1, A)$ are often called the elements of $A$. The reason for this can be explained as follows, if there is an adjuction $C\to Set$ that is a forgetful funtor that preserves products (atleast the empty one) has $A=Hom_{set}(*, A)=Hom_C(1, A)$. Now consider a collection of objects, $X_i$, in $C$ indexed by $I$, the map $\coprod_i X_i\to \coprod_i 1$ taking the unique final map. A section is a retraction of this map. In this case it will, by definition of coproduct, consist of a sequence of elements of each $X_i$, in the above sense. But this is exactly the data needed to give a map into the product of the $X_i$, equivalently an element. This generalizes your observation in the case that $C=Set$.
Other than that I'm not sure what direction you want?
I don't think this generalizes much in the direction of categories, atleast not in the naive sense. Case and point, a collection of objects indexed by $I$ is equivalent to a diagram $I\to C$ where $I$ is viewed as a discrete category. The product is the limit of this diagram and the coproduct the colimit, so lets start working with an arbitary indexing category $D$. Now, the constant diagram of final objects is also final in the category $Hom_{cat}(D, C)$, so we get an induced diagram as above, which induces a morphism of the colimits of these diagrams. To fix notation take $f:D\to C$ and $1:D\to C$ an arbitary diagram and the final one respectivly. The properties of the final object imply $colim(1)=\coprod_{\pi_0(D)}1$, and since we already dealt with the disconnected case, lets assume that $\pi_0(D)=0$ for now. Then we have that sections of $colim(f)\to 1$ are the same as elements of $colim(f)$. This is almost never in bijection with elements of $lim(f)$.
Though we can go in a different direction. To start off take $I$ an index set, $X_i, Y_i$ two collections of objects with morphisms $X_i\to Y_i$. Then sections are clearly in natural bijection with the product of sections of the original maps $X_i\to Y_i$. This actually fits in as a trival example of solving a lifting problem by reducing, which is useful when you have a filtration of your object and can inductivly lift by getting an element in each "obstuction group". Namely, if we have a filtration $F_i$ of an object $X$, and we are trying to construct a section $Y\to X$, we may first try to construct it on each $F_i$ inductivly. The tragic tale about this though, is that instead of a product, you ussally get a spectral sequence, which degenerates in the trivial case. Key examples are the Homotopy spectral sequeunce and the Bousfield-Kan spectral sequeunce which work well on simplicial sets, though this is a bit far.