Alternative closed structure on Cat

[Ondrej Rypacek <ondrej.rypacek@gmail.com>]

Dear All

Is there a name for and what is known about the tensor product of ordinary categories which looks like the underlying 1-category of Gray's (Gray) tensor product? 
Explicitly, roughly: 
- objects of C \otimes D are pairs (c,d) , c object of C , d an object of D
- arrows alternating lists of arrows from C and D, i.e. they are generated by 
 	(f,d) : (c,d) -> (c',d) for f : c -> c',
 	(c,g) : (c,d) -> (c,d') for g : d -> d'

and modulo the equations:   (f',d) . (f, d) = (f'f, d), (c,g') . (c,g) = (c,g'g), and identities, left and right unit laws and associativity in each component separately.
	

And is the category of categories with respect to this tensor closed ? 


Thank you!
Ondrej



[For admin and other information see: http://www.mta.ca/~cat-dist/ ]