In an attempt to understand dual spaces and adjoints (in linear algebra) I came accross this video, which mentions "natural isomorphisms". Not knowing what a natural isomorphism is, I tried to look it up, and almost everything I found talked about category theory, or like in here claimed that a natural isomorphism is one that does not depend on the choice of a basis.
This made absolutely no sense to me. An isomorphism is a bijective linear function from a vector space to another. It takes a vector and spits another vector. How is a basis related to this anyhow?
So then I found this unanswered question before writing this, where one of the comment says that what depends of the basis is the construction of the isomorphism. But what does it mean for the construction of something to depend on a basis? Can a rigorous definition be given without needing to learn a lot of category theory?
Yes, it is the construction (i.e. the actual formal definition) of the linear map that depends on the basis.
There is an important thing lacking from the context here, namely that in the situations when such a question arises, we want to define this mapping (isomorphism) for all vector spaces at once, basically by the same formula.
(And here is the place where category theory turns into the picture: consider all (finite dimensional) vector spaces as vertices of a big graph, and consider linear maps as edges between them. The extra thing that makes it a category is composition of consecutive edges.)
The classical examples are the isomorphisms $\varphi:X\to X^*$ and $\psi:X\to X^{**}$ for finite dimensional vector spaces $X$, where the construction of $\varphi$ depends on a choice of basis (but besides that / after that, it uses the same formula for every $X$), whereas the construction of $\psi$ does not.