Monoidal structure on graphs

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

Greetings, fellow categorists.

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
everywhere) :

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

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-->g

forall  k: x-->y in X, p_1(k) : f(x)-->g(y)

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

I would be quite surprised if this structure had never been seen before.
Enriched universal algebra in there has applications in computer
science.

Thanks,

Francois Lamarche