The category FNS of fuzzy neighbourhood spaces and their continuous functions is closed under the formation of optimal and co-optimal lifts in the larger topological category FTS of fuzzy topological spaces. But, while the fuzzy neighbourhood system and the level topologies of an optimal lift of a source in FNS have simple descriptions in terms of the fuzzy neighbourhood systems and the level topologies, respectively, of the source spaces, analogous descriptions are yet to be given for both constructs for a co-optimal lift of a sink in FNS. We here establish such descriptions, with the help of a functor, introduced earlier by one of us, with codomain FNS and domain another full subcategory of FTS, the category FNBS of fuzzy neighbourhood-base spaces.