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The brown:green (really, carbon:nitrogen) ratio of your coop cleanings will depends HUGELY on how you manage your coop sanitation.
If your coop cleanings consist only of bedding-free scrapings from a droppings board, then it is very high nitrogen and will indeed need some shredded paper or straw or used shavings or whatever added to it to compost well.
However if you follow the common plan of cleaning all the bedding+poo out of the coop every week or two, and use a reasonable depth of bedding, and do not have your chickens crowded severely, then you will be having quite a lot more carbon than nitrogen, relative to what composts well; and you would want to add NOT another brown material such as shredded paper, but rather more of a so-called green material such as pure poo, fresh grass clippings or fresh veg/fruit scraps, urine, ammonia, high-nitrogen commercial fertilizer, or such.
I would suggest, particular with respect to processing coop cleanings, that you regard this brown:green ratio business as a starting place for experimentation, not an actual guideline, because it is fantastically difficult to correctly guess what the REAL ratio of C:N is in your particular coop cleanings. So, try it, see what happens, then adapt
Good luck, ahve fun,
Pat