Reopened, Re-closed (by a mod), and Reopened again with context edit reinstated
Please consider reopening Any criteria for a category to have all connected limits?. This is a very natural question about whether there is an analogue of certain well-known basic theorems in category theory in a less well-known context. The context in the original question was quite clear to anyone familiar with the area but was only in a brief parenthetical at the end of the question. I have expanded on this and improved the presentation a bit to make it more accessible to non-experts.
It seems to me that this is exactly the sort of question that this site was built for: questions that may naturally occur to many people (not just some exercise that is of no particular interest), do not have obvious or "standard" answers that can be found in every textbook on the subject, and by having an answer here the answer will be easily findable with a search engine.
[Disclosure: I have answered the question.]