Monoidal structure, take II

[Francois Lamarche <Francois.Lamarche@loria.fr>]

Given the private replies I got to yesterday's queries, it is obvious I
was not clear enough, and indeed there was an unhelpful typo.

> 
> I'm wondering, if anybody has ever described the following monoidal
> structure on the category of oriented multigraphs, what MacLane calls
> graphs, the most common kind of graph in category theory (but not

OK Saunders, from now on they're graphs. This what happens when you hang
out with combinatorists AND category theorists.

> 
> Given graphs X, Y, the set |X-oY| of vertices on  X-oY  is the set of graph
> morphisms
> X --> Y.

So right from the start this is not the usual presheaf CC structure,
where the set of vertices is the set of all functions |X| --> |Y| .

So in what follows I use categorical notation for vertices, arrows, etc.
 
> Given f,g : X --> Y the set of arrows f-->g is the set of pairs
> (p_0,p_1) of functions such that
> 
> forall  x in |X|, p_0(x) : f(x)-->g(x)
> 
> forall  k: x-->y in X, p_1(k) : f(x)-->g(y)

Now the typo has been corrected. So an arrow f --> g is like a natural
transformation, with p_0 the usual familly of arrows indexed by the
vertices/objects of X, but since things don't compose, you add the
diagonal p_1 as part of the information. There is some kinship to
homotopies, as M. Barr has remarked.

> This co-contra bifunctor has a tensor left adjoint, which is symmetric
> and monoidal.
> 

Is this more understandable?

Thanks again

Francois