One more comment on that fascinating ugly monoidal structure. Many years ago D. Pataraia as a student was asked to realise tensor product of vector spaces (V and W over k) as a colimit. He then came up with a diagram (sorry for still more ugly notation) k_{v,w} / \ / \ |_ _| V_w W_v That is, vertices of the diagram consist of U(W) copies of V, U(V) copies of W, and U(V)xU(W) copies of k (U is the forgetful functor to sets). And the maps... well, you guess. The reason this is relevant is that in the Barr's monoidal category, the product of (S->U(A)) and (T->U(B)) is (SxT->U(C)) where C is the colimit, in the category of algebras, of F(1)_{s,t} / \ / \ |_ _| A_t B_s It does not look so ugly after all, does it? :), Mamuka