We show that a commutative monoid A is coexponentiable in CMon(V) if and only if $-\otimes A : V \to V$ has a left adjoint, when V is a cocomplete symmetric monoidal closed category with finite biproducts and in which every object is a quotient of a free. Using a general characterization of the latter, we show that an algebra over a rig or ring R is coexponentiable if and only if it is finitely generated and projective as an R-module. Omitting the finiteness condition, the same result (and proof) is obtained for algebras over a quantale.
Keywords: monoidal category, projective module, rig
2010 MSC: 18A40, 18D15, 13C10, 16Y60
Theory and Applications of Categories, Vol. 32, 2017, No. 36, pp 1222-1228.