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 closed embedding if the corresponding algebra map $\phi :B' \to A$ is surjective.

It is easy to see that when $\phi$ is surjective then for any $R$ -$k$ algbera the homomorphism $\psi_R : H'(R) \to G(R)$ is injective. But I am not sure about the converse.

Natural definition of Embedding could have been the homomorphism $\psi_R : H'(R) \to G(R)$ is injective for $R$ -$k$ algbera but the author has defined otherwise. Is there any example where each $\Psi_R$ is injective but the corresponding map $B' \to A$ 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.