Luc seems to be describing what, in our work on rewrites etc, Eilenberg wanted to call a one-and-a-half-category, and I changed (and I'm not sure how Sammy feels about it) to sesquicategory when writing the article "Categorical structures" for "Handbook of Algebra" Volume 2 (Elsevier, North Holland). [The preprint is dated Nov 1992 but the volume probably won't appear until late 1994!] There are two symmetric monoidal closed structures on the category Cat of categories. One is the usual cartesian closed structure where the internal hom [A,B] is the usual category of functors A --> B and NATURAL transformations between these. The other closed structure (which I call the "funny" one) has internal hom [[A,B]] the category of functors and transformations between these (which consist of the data for a nat trans without the naturality requirement). This funny structure WAS an example in Eil-Kelly "Closed categories" La Jolla 1965. Carolyn Brown has used the funny structure in her work on Petri nets. Sesquicategories are V-categories where V is Cat with the funny monoidal structure. John G. Stell <john@cs.keele.ac.uk> tells me he has used sesquicategories in computer science, and apparently independently came up with the same name for them. Perhaps of future interest to those interested in sesquicats are the various monoidal structures on 2-Cat. For example, there is John Gray's tensor product of 2-categories, where "natural", instead of being dropped, is replaced by "lax natural". A recent paper of Gordon, Power, Street looks at the case where V = 2-Cat with the monoidal structure obtained by replacing "natural" by "pseudonatural"; we prove that every tricategory is equivalent (in the approp sense) to a V-category. Regards, Ross ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++