Characterize test Morita equivalences via Set-valued models

Establish that for a morphism of limit sketches F: S -> T, the following are equivalent: (i) F is a test Morita equivalence, meaning that for every test sketch M (a normal sketch whose underlying category is complete and cocomplete with all small limit and colimit diagrams specified), the precomposition functor F^*: Skt(T, M) -> Skt(S, M) is an equivalence of categories; and (ii) the precomposition functor F^*: Skt(T, Set) -> Skt(S, Set) is an equivalence of categories, where Set is the category of sets equipped with its standard test sketch structure.

Background

In the paper, the authors introduce test sketches (normal sketches with complete and cocomplete underlying categories and all small limits and colimits specified) and define a test Morita equivalence between sketches S and T as a morphism F: S -> T such that for every test sketch M, precomposition F^*: Skt(T, M) -> Skt(S, M) is an equivalence. This captures the idea that S and T have the same models in all test sketches.

They conjecture that, for limit sketches, it suffices to check equivalence of models only in Set with its natural test structure. The authors attempted to prove this but were unable to do so, citing delicate size issues, and suggest that restricting test sketches to large algebraic finitely accessible (LAFT) categories might circumvent these obstacles.