A locally modeled monoid or, R-Ring, is a generalized quantity in the sense of space and quantity which is modeled on a category of local models R.
Let (R,U,L,A) be a category of local models. Then an R-ring or monoid locally modeled on R is a co-presheaf
R \to Set
which preserves the limits of shape in L.
A morphism of such locally modeled monoids is a natural transformation.
This is definition 1.1.6 of
where it appears as part of the discussion of derived smooth manifolds.