Carboni and Walters (Cartesian bicategories I, JPAA 49(1987), p 13) state: "Consider a bicategory B with a tensor product. Then the tensor product is the biproduct iff every object has a cocommutative comonoid structure and every arrow is a comonoid homomorphism." Presumably this fact should specialize to the case of symmetric strict monoidal categories, so that for a symmetric strict monoidal category C, the tensor product is the cartesian product iff every object has a cocommutative comonoid structure and every arrow is a comonoid homomorphism. Let C be a symmetric strict monoidal category, with identity object I, tensor product denoted by *, and with symmetry isomorphisms sXY: X*Y --> Y*X. Suppose each object X in C has a cocommutative comonoid structure given by the arrows: tX: X --> I dX: X --> X*X The comonoid equations are (dropping the names of the objects for simplicity): (1*t)d = 1 (counit) (t*1)d = 1 (d*1)d = (1*d)d (coassociativity) sd = d (cocommutativity) Suppose further that every arrow of C is a comonoid homomorphism. It should then follow that the tensor product is the cartesian product. Presumably the projections would be (1*t): X*Y-->X and (t*1): X*Y-->Y, and the target tupling of f: Z-->X and g: Z-->Y would be given by (f*g)d. The equations that would have to be satisfied are: (1X*tY)(f*g)dZ = f (tX*1Y)(f*g)dZ = g (((1X*tY)h)*((tX*1Y)h))dZ = h The first two are easy to prove. Can someone tell me how to prove the third? I am probably missing something simple, but I spent a few hours on it and I just don't see it. I can reduce it to the problem of establishing either ((1X*tY)*(tX*1Y))d = 1 or else d(X*Y) = (1X * sXY * 1Y)(dX * dY) but I don't see how to prove either of these from the stated assumptions. - Gene Stark ==================================