In the course of reading Mac Lane linearly (currently in Chapter VI), I have seen again and again that duality can make life much easier. My problem is that I have almost no background in logic, and duality is a theorem in logic about category theory.
When I first read about duality in Chapter II of Mac Lane in the context of the elementary theory of a single category, everything was pretty clear even without knowing any logic. However, when I got to the chapter on adjunctions, involving two categories and functors between them, a bijection of hom-sets, and two natural transformations, I got confused to the point that I wasn't even sure how to use duality (let alone, why it is correct).
At this stage, I made a rather long pause and read the first three chapters of Ebbinghaus, Flum, and Thomas' ''Mathematical logic'' (so, I have read about the syntax and semantics of first-order logic). From this, I built my own (hopefully correct) ''poor man's proof of duality'' up to the situation of a single adjunction. This has both clarified the validity of duality for formulas involving adjunctions, and helped me understand how to use duality in such situations.
But a single adjunction is far from the most ''complicated'' situation one meets. There are composition of adjunctions, pointwise limits in functor categories, and many other situations in which I am still not totally convinced that I understand duality (both theoretically and practically).
For example, in one answer to a recent question on pointwise limits in functor categories, it was stated that the reference for limits is Mac Lane, while the reference for colimits is Mac Lane--Moerdijk. I really wanted to comment that the assertion on colimits is just the dual of the one on limits, but then I realized that I am not totally sure. I would be most grateful for some solid source that I can consult whenever I have doubts in what I get after doing the intuitive things (reverse arrows but not functors, etc.).
- What is a good reference for an explicit, logic-based, statement of a duality theorem of category theory in ''complicated situations?''
- What are the prerequisites in logic? For example, up to which point of Ebbinghaus--Flum--Thomas should I read?