For every in , the evaluation at , that is, the map formula_23 defines an algebra homomorphism from to , which is the unique homomorphism from to that fixes , and maps to . In other words, has the following universal property. "For every ring containing , and every element of , there is a unique algebra homomorphism from" "to that fixes , and maps to ." As for all universal properties, this defines the pair up to a unique isomorphism, and can therefore be taken as a definition of .