Dear Steve, At 09:54 PM 12/2/2010, you wrote: If I understand correctly, you have arrows f:X->Y and g:Y->Z and you are comparing the tensor products f@g:X@Y->Y@Z and g@f:Y@X->Z@Y. They have different domain and codomain, so cannot be equal. I was thinking more about f:X->Y and g:Y->Z and tensoring f with the identity morphism of Y to get f@1_Y:X@Y->Y@Y, and also tensoring 1_Y with g to get 1_Y@g:Y@Y->Y@Z. So I get the composition f@1_Y followed by 1_Y@g is a morphism from X@Y->Y@Z, and you made me realize that f followed by g as a morphism X->Y cannot possibly equal f@1_Y followed by 1_Y@g. Then again, if the ambient strict monoidal category is symmetric, so that the latter composition is a morphism X@Y->Z@Y, then to my mind somehow this is pretty much the same as the composition X->Z of f followed by g, basically because only the identity morphism of Y is involved. The context of my inquiry is chemical reaction, as suggested by John Baez a while ago. That is, if f and g are chemical reactions that transform X to Y and Y to Z, respectively, then the net effect is just transformation of X to Y, since the Y produced by f is completely consumed by g. Bottom line: I would like a correct way to say that tensoring f with an identity morphism is somehow no different from f. Ellis [For admin and other information see: http://www.mta.ca/~cat-dist/ ]