Measuring coalgebra ($p(c)(ab)=\sum p(c_{(1)})(a)p(c_{(2)})(b)$)

For unital algebras over a field $A,B$, a coalgebra $C$ measures $A$ into $B$ when a linear map $p:C\to {\mathrm{Hom}}_k(A,B)$ satisfies the displayed identity and $p(c)(1_A)=\epsilon (c)1_B$. Group-like elements therefore act by unital algebra homomorphisms over a field.

A terminal measuring coalgebra: every measuring map from $C$ factors uniquely through its universal measuring map by a coalgebra homomorphism $C\to P(A,B)$. Composition of universal measurings makes $P(A,A)$ a bialgebra, with the identity map providing its unit.