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Where does this concept come from? The page lists no relevant references, nor can I find any search results for “quadrable cospan”. Furthermore, the “Note on terminology” mentions the terminology “carrable”, which means something different as far as I can tell (and certainly in the cited references): namely, a morphism along which all pullbacks are admitted.
Just to say that I don’t know (I only added the diagrams that I felt were missing), the original statement in revision 1 was by user “Harry”, who, if I recall correctly, was Harry Gindi. (Yes, the original discussion for this entry is here)
Harry hasn’t been active here in a long time and may be unlikely to see your question.
But since the term “quadrable” is unlikely to be used much (Google search suggests the only place where it is mentioned is at cospan) there would probably be little harm in clearing or else re-naming/wording the entry.
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