lo.logic - What is a reference for an explicit, logic-based, statement of duality in category theory (in ''complicated'' situations)? And what are the prerequisites for a beginner in logic?

Background

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.).

Questions

1. What is a good reference for an explicit, logic-based, statement of a duality
   theorem of category theory in ''complicated situations?''


2. What are the prerequisites in logic? For example, up to which
   point of Ebbinghaus--Flum--Thomas should I read?