TITLE: 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?
QUESTION [6 upvotes]: 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

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?

REPLY [9 votes]: I would go even farther than the comments above, at least in the specific case you mention about computing (co)limits objectwise in a functor category.  Once you know the statement for limits, deducing the statement for colimits is not even a syntactic transformation to the proof, and needs no arguments from formal logic at all.  Instead, to prove that colimits are computed objectwise in the functor category [I, C], simply use the fact that limits are computed objectwise in the functor category [Iop, Cop] = [I, C]op, and the fact that colimits in a category are the same as limits in the opposite category.
I think this is the common case in this kind of argument, but perhaps someone can come up with an example where the argument really needs to be repeated in the dual situation.