I'm trying to think of how posets simplify the conditions of the general https://ncatlab.org/nlab/show/adjoint+functor+theorem. First, it is clear that limits and colimits in posets are only meets and joins, because diagram commutation is nothing more than the existence of "morphisms", i.e. relation elements. And of course, for this same reason posets are locally small. At first glance, I'm not sure how one might verify the "solution set condition" here; and I'm not familiar enough with "cototal" to judge. The other criteria is codomain being "well-powered", i.e. every object has a small poset of subobjects, this is trivially true; but lastly, I am not sure about "cogenerators", because the classic cogenerator is the subobject classifier, but Poset does not have one: https://math.stackexchange.com/questions/1650277/subobject-classifier-for-partial-orders/1650434. So, I've brought up a bunch of fancy stuff without a conclusion! But if anyone has any insight on this, it would be appreciated. (Ah, I think the "solution set" and "small cogenerating set" are trivially true if the posets are small... are we assuming that?)