The solution set condition appears as part of the hypothesis in Freyd’s General Adjoint Functor Theorem.
A functor satisfies the solution set condition if for every object of there exists a small set , an -indexed family of objects of and an -indexed family of morphisms in such that each morphism in can be factored, for some index and some morphism in , as
This is a smallness condition in that the family is required to be indexed by a small set.
A restatement of this condition is that the comma categories all admit weakly initial families of objects.
Here is the connection with the adjoint functor theorem: when small products exist in those comma categories, this is equivalent to saying that they all admit weakly initial objects. When small equalizers exist in those comma categories also, this is equivalent to saying that they all admit initial objects, and this is equivalent to being a right adjoint.
To define the projective tensor product of Banach spaces using the adjoint functor theorem, let and each be Ban (the category of Banach spaces and short linear maps), fix a Banach space , and let be , the Banach space of bounded linear maps from to .
If is a short linear map and is a bounded linear map, then is a bounded linear map with norm , which proves that , so that is also a short map, as it must be for to be a functor. (That is linear and preserves identities and composition is trivial.)
The solution set condition now states: For every Banach space there exists a small set , an -indexed family of Banach spaces and an -indexed family of short linear maps such that each short linear map can be factored, for some index and some short linear map , as .
(to be completed)