Measuring coalgebra

In algebra, a measuring coalgebra of two algebras "A" and "B" is a coalgebra enrichment of the set of homomorphisms from "A" to "B". In other words, if coalgebras are thought of as a sort of linear analogue of sets, then the measuring coalgebra is a sort of linear analogue of the set of homomorphisms from "A" to "B". In particular its group-like elements are (essentially) the homomorphisms from "A" to "B". Measuring coalgebras were introduced by .
Definition.
A coalgebra "C" with a linear map from "C"×"A" to "B" is said to measure "A" to "B" if it preserves the algebra product and identity (in the coalgebra sense). If we think of the elements of "C" as linear maps from "A" to "B", this means that "c"("a"1"a"2) = Σ"c"1("a"1)"c"2("a"2) where Σ"c"1⊗"c"2 is the coproduct of "c", and "c" multiplies identities by the counit of "c". In particular if "c" is grouplike this just states that "c" is a homomorphism from "A" to "B". A measuring coalgebra is a universal coalgebra that measures "A" to "B" in the sense that any coalgebra that measures "A" to "B" can be mapped to it in a unique natural way.