Defining adjoint functors: What does "natural bijection" mean?

Question

Take the following definition of adjunction from the nlab

This definition can be found in numerous other places. My brain parses this definition perfectly up till the point where it says that for every $c$ and $d$ the hom-sets are naturally isomorphic. At his point it does SCRREEETCH, because, as a beginner, learning category theory, natural isomorphisms were only defined between functors as a collection of morphisms in the target category and not *between collections of morphisms lying in different categories.

I know that there are other equivalent definitions of adjunctions, but I don't have the time to go through them to see how to make sense of this definition. I just want a clean explanation how to understand this definition.

Answer 1

Converting my comment to an answer, as suggested:

$\text{Hom}_D(L(c),d)$ and $\text{Hom}_C(c,R(d))$ are both sets, i.e. they're objects of the category $\mathsf{Set}$, which is the target category where the natural isomorphism is taking place.

You wrote:

My brain parses this definition perfectly up till the point where it says that for every c and d the hom-sets are naturally isomorphic.

It's not that the $\text{Hom}$ sets are naturally isomorphic - there's just a bijection between them, and these bijections cohere to a natural ismorphism between the $\text{Hom}$ functors.