Source: https://en.wikipedia.org/wiki/Seifert%E2%80%93van_Kampen_theorem
I am aware of "natural transformations" in category theory. However, I have trouble figuring out in this context why is $k$ called a "natural morphism"?
I can't figure out which are the "functors" and the "objects" in this case, to qualify $k$ as a natural transformation.
Thanks a lot.
$k$ is a natural transformation, although it's annoying to spell out in full detail how. The relevant functors have domain the category of tuples consisting of a topological space $X$, two open subspaces $U_1, U_2$ of $X$ with nonempty intersection, and a basepoint $x_0 \in U_1 \cap U_2$ (I'll leave the description of the morphisms in this category as an exercise). The codomain is groups. The first functor is $\pi_1(U_1) \ast_{\pi_1(U_1 \cap U_2)} \pi_1(U_2)$. And the second functor is $\pi_1(X)$ ($x_0$ is the basepoint throughout).
But as in the comments, "natural" here is also being used in a somewhat more informal sense to refer to a map that exists for "universal reasons" in some sense, by which I mean because of universal properties. Namely:
The spaces $U_1, U_1 \cap U_2, U_2, X$ fit into a commutative square (in fact a pullback square), and by the universal property of pushouts, this induces a map $U_1 \sqcup_{(U_1 \cap U_2)} U_2 \to X$ which preserves basepoints. Applying $\pi_1$ to this map (and using the fact that functors $F$ always have the property that there is a natural map from $\text{colim} F(-)$ to $F(\text{colim}(-))$, again by the universal property of colimits) produces the desired map $k$.
A simpler example of a natural map in this sense, although it is also a natural transformation, is the fact that $U_1 \cap U_2$ admits two natural inclusions, one into $U_1$ and one into $U_2$, by the universal property of intersections.
It's actually fairly unfortunate that "natural" gets overloaded in this way because this second meaning is subtle and takes some getting used to, but this is standard usage in category theory. Maybe we should switch to "universal" instead or something.