Dear Ondrej and Peter The fact to which Peter referred, that the tensor product in question
is the unique other symmetric monoidal closed structure on Cat
was proved in the paper [1] F. Foltz, G.M.Kelly, and C. Lair, "Algebraic categories with few biclosed monoidal structures or none", JPAA 17:171-177, 1980 As for the name, this tensor product has been called the "funny tensor product" by some authors. But as I argued in my paper [2] Free products of higher operad algebras http://arxiv.org/abs/0909.4722 in which such a tensor product is defined for any structure definable by a "normalised higher operad" in the sense of Batanin, the name "free product" is a better choice of terminology. Mark Weber [For admin and other information see: http://www.mta.ca/~cat-dist/ ]