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