From: Peter Freyd [Let *D* be the category whose objects are] set-valued bifunctors on *A* (contravariant on the first variable, covariant on the second) Two definitional/notational suggestions: 1. "sesquifunctors on A" for the objects of *D* 2. A -X A as a notation for *D* The right-hand half of the X in -X is intended to suggest the contravariant bit. One then has the following hierarchy. A => B "functions" from A to B, viz. !A -o B A -o B (homo)morphisms from A to B A -X B sesquifunctors A\op x A -> B, dinaturally transformed The first two come from linear logic, with !A = FU being interpreted as the underlying object (e.g. underlying set) of A reflected back into the category via F. F serves merely to make the underlying objects the same "kind" as the objects they underlie, avoiding a proliferation of kinds in order to keep linear logic typeless, or at least kindless. Vaughan Pratt