I read this post and didn't come away with much.
Here is an example scenario: consider the collection of all categories possessing finite coproducts and a terminal object. I would like to prove the existence of a category I such that there is exactly one isomorphism class of coproduct preserving and terminal object preserving functors from I to any other category with finite coproducts and a terminal object. I suspect that the category of finite sets of natural numbers and all functions between them is a decent model of I, and I (think I) worked out the proof of this.
I want to find out if I is (1) unique up to equivalence of categories and (2) is equivalent to the topos of finite sets (since my supposed model of I is)
I don't know how to prove (1) or (2), and wading through the nLab is making my head spin.
Can somebody please walk me through this example? Can somebody please point me toward constructions of similar initial categories possessing other universal properties (for simplicity, expressed as adjunctions in CAT)?