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As written, the Idea section doesn’t seem to have much content that is specific to algebraic compactness, just very general remarks about data structures being fixed points.
As a minimum fix I have added the following lead-in to the Idea section (hoping that it will prompt some expert to expand/improve on it):
A category is called algebraically compact if for every endofunctor on it the respective initial algebra coincides with the final coalgebra.
Under categorical semantics of programming languages this condition ensures the existence of inductive-recursive types (e.g. Zamdzhiev 20). For that, recall:
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