I thought I had invented bicategories in 1967, and that, at the very beginning of the paper, in §1, I had defined the two composition laws and drawn pictures to explain them. Of course I denoted by capital letters the 1-cells, thinking of functors, and by small letters the 2-cells, thinking of natural transformations. That certainly makes a tremendous difference with Susan Niefield's notation who uses the converse convention and amply justifies Marco Grandis in giving references dated 1994 and 1996, i.e. more than 25 years posterior to my original paper. With best regards

You can find the strict version of that result in Prop. 1.4 of

- M. Grandis, Homotopical algebra in homotopical categories, Appl. Categ. Structures 2 (1994), 351-406.

I do not know if it has been written down elsewhere.

For sure, whiskering of natural transformations with functors is used in:

- R. Street, Categorical structures, in: Handbook of Algebra, Vol. 1, 529-577, North Holland, Amsterdam 1996.

where you can find the notion of a sesqui-category (which does not assume the "reduced interchange axiom" you are mentioning).

With best regards

M. Grandis

...

Does anyone know of a reference for the following definition of a bicategory? The primitive composites are:

gf for composable 1-cells GF for vertically composable 2-cells f*G and F*g for horizontally composable pairs of each

with appropriate axioms including (G*f')(g*F)=(g'*F)(G*f), for F:f->f':X->Y and G:g->g':Y->Z. The horizontal composite G*F is defined to be the common value of the two vertical composites.

-Susan