Inductive limits commute "naturally" with binary products in Set I am taking an introductory course in Category Theory, and one of the problems is

Prove that inductive limits commute with binary products in Set; i.e. for infinite sequences of sets $\{X_n\}_{n \in \mathbb{N}}$ and $\{Y_n\}_{n \in \mathbb{N}}$ with maps $X_n \to X_{n+1}$ and $Y_n \to Y_{n+1}$ construct a natural map
$$
\text{colim}_n(X_n \times Y_n) \to \text{colim}_n(X_n) \times \text{colim}_n(Y_n)
$$
and show it is an isomorphism.

Constructing the map
We may view $\text{colim}_n(X_n)$ as the disjoint union $\sqcup_{n}X_n$, quotiented by the equivalence relation $\sim$, where $x \sim x'$ if and only if $x$ and $x'$ are eventually mapped to the same element. We may view the other colimits similarly.
Then define the map $\Phi:\text{colim}_n(X_n \times Y_n) \to \text{colim}_n(X_n) \times \text{colim}_n(Y_n)$ by $\Phi([(x_n, y_n)]) = ([x_n],[y_n])$, where the square brackets denote cosets of sequences under the respective equivalence relations. It is easy to see that this map is well-defined.
We can also show quite quickly that $\Phi$ is a bijection, hence an isomorphism in Set.
My problem
I don't know what is meant by saying that the isomorphism has to be a natural map. The only definition of natural map I have been given is that of a natural transformation $\eta:F \implies G$ between functors $F, G$, and functors take a single object as an argument. In this case, it seems that naturally must refer to "naturality in $X_n$ and $Y_n$" in some sense, but there are infinitely many of each. What is actually meant by naturality of the map?
 A: There is a category of pairs of sequences $X_1 \to X_2 \to \dots, Y_1 \to Y_2 \to \dots$ of objects in $\text{Set}$. This category admits two functors to $\text{Set}$ given by the LHS and the RHS respectively, and the isomorphism you want to write down is a natural transformation between them.
This is not what is meant by "natural," though. What is meant by "natural" here is that this natural transformation arises in a "universal way," as follows. If $F : C \to D$ is any functor whatsoever, consider a colimit $\text{colim}_j \, c_j$ of objects in $C$ such that the colimit $\text{colim}_j \, F(c_j)$ in $D$ exists. Then there is a particularly canonical map
$$\text{colim}_j F(c_j) \to F(\text{colim}_j \, c_j)$$
which is determined by the universal property of the colimit to be the map corresponding to $F$ applied to the inclusions $c_j \to \text{colim}_j \, c_j$. We say that $F$ preserves colimits if this particularly canonical map is an isomorphism; the exactly dual construction describes what it means for a functor to preserve limits.
A naive definition of "preserves colimits" might be that there is some natural isomorphism between the LHS and the RHS, but that isn't what "preserves colimits" means, it means that the particularly canonical map written down above is an isomorphism. Whether it's possible to formalize what "canonical" means here is genuinely unclear to me; I asked about this on MathOverflow here.
It is a very unfortunate terminological fact about category theory that "natural" has at least three different meanings, one of which is informal: sometimes it means functorial, sometimes it means natural-as-in-transformation, and sometimes it means this third thing about canonical maps I don't know how to formalize.
