$\DeclareMathOperator\Hom{Hom}$I am reading Introduction to affine group schemes by Waterhouse. In the second chapter he defines a closed embedding in the following way. Let $G = \Hom_k(A,-)$ and $H'= \Hom_k(B',-)$ be affine group schemes and $\Psi : H' \to G $ be a homomorphism. $\Psi$ is called a closed embedding if the corresponding algebra map $\phi :A \to B'$ is surjective.
It is easy to see that when $\phi$ is surjective then for any $k$-algebra $R$ the homomorphism $\Psi_R : H'(R) \to G(R)$ is injective. But I am not sure about the converse.
A natural definition of "embedding" could have been the homomorphism $\Psi_R : H'(R) \to G(R)$ is injective for all $k$-algebras $R$ but the author has defined otherwise. Is there any example where each $\Psi_R$ is injective but the corresponding map $A \to B'$ is not surjective? May be this is the reason for the definition given in Waterhouse?
Note: Waterhouse does not consider $A$, $B'$ to be finitely generated $k$-algebras.
