Add to ORKGWe examine a variety of logical framework of partial algebras, of sketches and of order sorted theories with sort constraints. Therefore we use weak embeddings of institutions which preserve model categories and may map sentences to theory extensions using unique-existential quantifiers. They faithfully represent theories, model categories, theory morphisms, colimit of theories, reducts etc. Our main result states the equivalence of all the logical frameworks with respect to weak embeddability. This gives us compilers between all frameworks. Thus it is a chance to unify the different branches of specification of partial algebras. Further, one can combine specifications written in different logical frameworks by translating them in one's fav...