# Some lengthy question on natural transformations, category theory, and dual objects

So I am fairly new to algebra, and my instructor often uses terms from category theory (such as Hom-set, natural transformations, etc) that I am not familiar with. As an attempt to have a better understanding on the subject, I post this question.

Question 1: There are "morphisms" which my textbook(Herstein)define as "functions" (with some specific properties). But I have a feeling that this definition is somewhat inadequate in that morphisms actually include functions (morphisms between sets).

So how does one define PRECISELY (in the language of set theory) a morphism?

Question 2: So there are things called "natural transformations" that appear quite frequently in my problem sets. How does one determine PRECISELY when an isomorphism is natural? For instance, I know that a finite dimensional vector space $V$ is isomorphic to its dual, $V^*$, but this isomorphism is not natural in the sense that one must choose an "unnatural" basis to construct such isomorphism. Yet $V\cong V^{**}$ natural, where $V^{**}$ is the double dual. But it seems that this kind of "definition" is somewhat of an ad hoc, as there are many isomorphisms do not involve the concept of basis...

Question 3: So I mentioned it already, the dual vector space, the set of all linear functionals. But how does one construct such "dual objects"? I've seen may co-stuff so far: codimension, cokernel, coproduct, etc. Is there a precise and UNIVERSAL definition one can apply to get these dual objects?

2) The precise definition of a natural transformation is something you should be able to find in any reasonable text on categories. A construction is natural when it fits into a natural transformation between some relevant functors, between some relevant categories. The double dual construction is naturally isomorphis to the identity functor (for finite dimensional vector spaces), and this can be proven by following the definitions. Any particular choice of an isomorphism between $V$ and $V^*$ can be shown to fail the condition of a natural transformation, and thus is not natural. Informally, this has to do with a dependency on a basis, but this is just a technicality of this particular case.