categories: Final Coalgebra Question

[N Ghani <ng13@mcs.le.ac.uk>]

Can anyone help me with the following final coalgebra questions

1. Let X be a fixed Set. What is the final coalgebra of the functor
[X,_]:Set -> Set. If you wish to make X finite then that's fine by me.

2. Consider the functor [[_,2],2]:Set -> Set. This functor doesnt have
a final coalgebra for cardinality reasons. However one may define a
finitary variant of this functor as follows:

First let TX = [[X,2],2] if X is finitely presentable. Thus T:Set_fp
-> Set is a functor from the full subcategory of fintely presentable
objects of Set into Set. Next define T' to be the left Kan extension
of T along the inclusion Set_fp -> Set. In other words T'X is the
filtered colimit of all the TX_0 where X_0 is a finitely presentable
subobject of X

Now, T' is clearly finitary and from general nonsense we know that it
has a final coalgebra. But what is it concretely?

Thanks for any help you can offer

Neil