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

Question

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?

Much thanks in advance

Answer 1

1) There is no definition of morphism. There is a definition of category, and a category consists of objects and morphisms. That is what morhpisms are. Just like there is no definition of what a vector is. There is a definition of vector space. So a vector is (precisely) anything which is an element of some vector space. Similarly, a morphism is (precisely) anything which is a morphism in a some category.

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.

3) There is a general notion of a dualizing object in a category, but this does not capture all cases of dualities in mathematics.

Answer 2

The other poster already gave a good answer explaining some of your questions, so I would just like to recommend that you pick up Aluffi's algebra text. It teaches algebra and category theory simultaneously from the very basics, and I personally love it. However, he takes a while to introduce things like natural transformations and adjunctions, and just glosses over the details on these topics, so I would suggest that you also get yourself a reference text on category theory to look through as well. Simmons is a very easy and gentle introduction, and covers most of the essentials. Awodey is a bit tougher, but in my opinion it is often more clear and insightful than Simmons, and it also covers more content.