Hello, I've identified some useful non-standard properties of lax monoidal functors on Cartesian-like monoidal categories, and I am curious as to whether these properties already have names. Suppose I have a monoidal category with diagonals (diag : A -> A * A). Some lax monoidal functors (F, merge: FA * FB -> F(A * B)) have the property that diag;merge : FA -> F(A * A) equals F(diag). Is there a name for this property for either the special Cartesian case or the general case? What about the 2-categorical case where there's a 2-cell from diag;merge to F(diag) or vice-versa? Now suppose I have a monoidal category with terminators (term : A -> I). Some lax monoidal functors (F, unit: I -> FI) have the property that term;unit : FA -> FI equals F(term). Again, is there a name for this for the Cartesian, general, or 2-categorical cases? While I'm at it, is there a name for Cartesian-like monoidal categories with diagonals, terminators, and projections? The best I've found is Cartesian structures on bicategories, but my monoidal categories do not have the required 2-cells/identities. At best, when there are 2-cells, the 2-cells are not isomorphisms, which I guess makes it a lax Cartesian structure. Thanks for any help you can provide, Ross P.S. This is my first time using this mailing list, so please tell me if there's anything I should know about. [For admin and other information see: http://www.mta.ca/~cat-dist/ ]