In the SEP article on non-well-founded set theories, they make the point that (up to isomorphism at least), working in a non-well-founded set theory such as $AZF$ yields no more expressive power over $ZF$. Specifically, I refer to the result that the same functors can be shown to have final coalgebras over both $AZF$ and $ZF$.
In other words, $AZF$ does not allow us to talk about coalgebraic structures that $ZF$ is too weak to talk about, but simply gives us a more convenient vantage point from which to talk about them formally.
My question is, does this phenomenon hold more generally (say, over a Cartesian closed category, perhaps with natural number object), given that initial algebras exist for a certain class of functors, can we also construct final coalgebras over the same class of functors?
For a particular instance of this (which I think is a very general research question which may not have been explored as of the writing of this question), this is related to my other question, to the extent that proof theoretic ordinals measure the "expressability" of theories: Are $M$ types more powerful than $W$ types?
I wonder whether or not this is the case in general (under whatever framework) because from what I have seen in the literature, all attempts to increase the power of type theories (in the sense of ordinal analysis anyway) have been by proposing more and more general induction schemes, and not the analogous coinduction schemes. Is there a reason for this, or have researchers simply not looked too deeply into the topic as of yet?