Fixpoint operators for domain equations (Q1605186)

We investigate fixpoint operators for domain equations. It is routine to verify that if every endofunctor on a category has an initial algebra, then one can construct a fixpoint operator from the category of endofunctors to the category. That construction does not lift routinely to enriched categories, using the usual enriched notion of initiality of an endofunctor. We show that by embedding the 2-category of small enriched categories into the 2-category of internal categories of a presheaf topos, we can recover the fixpoint construction elegantly. Also, we show that in the presence of cotensors, an enriched category allows the fixpoint construction.