20241008
This following is a very brief summary of what happened in class on 20241008.
We introduced three relationships that might exist between a pair of functors
 We could use
to send into , and then consider all arrows in  We could use
to send into , and then consider all arrows in
In other words, we could consider the two sets

Case 1:
and are mutual inverses
In this case we haveand , and both functors are said to be isomorphisms. We then say that the categories and are isomorphic. The functors and establish bijections both between the objects of each category, and the arrows of each category. The two categories are, for all intents and purposes, essentially the same. So here we definitely have a (somewhat trivial) bijection between the homsets considered above. 
Case 2:
and are "almost" inverses
In this case we are supposeand , i.e., the two compositions are natural isomorphic to the identity functors. This is more flexible than the notion of isomorphism, and the corresponding adjective for the categories and is to say they are equivalent. It turns out that this is essentially the same as saying that the two categories are "the same up to isomorphism classes of objects." In other words, there might not be a bijection on objects, but if we identified isomorphic objects in each category, the results would look the same. 
Case 3:
and are "related"
In this case we supposed that there was a natural bijection between the homsets we originally considered. In other words, for everyand there is a bijection We'll make the "naturality" precise next class.
We focused our attention on the last case, in which case we say the two functors are adjoints. We also call the functor
We noted that the natural bijection
We then briefly talked about an example, namely the free abelian group functor as being left adjoint to the corresponding forgetful functor.
We will talk more about adjoints next class!
Concepts
References
 Mac Lane: pages 7989