This is the first part of a work concernedwith the study
of the following type of structure: A family of categories S_(A,B) (A, B in a set So) together with pairing functors
c(A, B, C): S(A, B) • S_(B,C) -~ S_(A,C) which up to given coherent iso- morphisms behave as if the S_(A,B) were the Hom?(B, A) for some
"category" ?. The best known cases are perhaps So = one point, then
we have a single category S with a multiplication in the senseof [B. 1],
or a 2=category lB. 3] where the associativity isomorphisms are identities,
or S = a set of rings, S(A, B) = category of (A, B)-Bimodules and -o
c(A,B,C)= ~) .
B