In Steve Awodey's book Category Theory, he mentioned a proposition called formal duality at page 54. The statement goes as follows:
Proposition $3.1$ (formal duality) For any sentence $\Sigma$ in the language of category theory, if $\Sigma$ follows from the axioms for categories, then so does its dual $\Sigma^*:$ $$CT \implies \Sigma$$ implies $$CT \implies \Sigma^*$$
What are some good examples to illustrate the point here? The theorem seems a bit of abstract to me and I don't have concrete examples in top of my head.
Moreover, the statement sounds intuitive to me but how can we proof it?
Edited:
And in Awodey's book, he stated that if a statement holds for every category so does its dual. I think the statement itself does not deserve to be a proposition but a corollary because for me, it is just a quick application of what we get from proposition $3.1.$ But why he stated this statement as a proposition?
I'm going to sketch a proof of the proposition. First of all, we need to fix a formal definition of $$CT \implies \Sigma$$ By Gödel's completeness theorem we can consider this statement either semantically or syntactically. It's more convenient to use the later way. At this point you may fix a formal system which you prefer (for instance, I'd choose Hilbert-style predicate calculus). Eventually, simple proof by induction on the height of a derivation of $CT \implies \Sigma$ gives us the desired result. In a base case you should realize that the operation $(-)^*$ preserves formulas in $CT$ (that is, sends $CT$-formulas into $CT$-provable formulas) and the axioms of your formal system. In an induction step you should prove $(-)^*$ doesn't affect applications of the rules of inference.
There are a great amount of examples of usage of duality result. You may encounter them almost everywhere in category theory. For example, if you proved some statement about limits, you don't need to prove the dual statement about colimits. To my opinion, it would be better, if you found the examples by yourself, because there is no some generic example, which would teach you all about duality.