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 Sweedler (1968, 1969).
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(a1a2) = Σc1(a1)c2(a2) where Σc1⊗c2 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.