Equivalence of Definitions of Unital Associative Commutative Algebra/Homomorphisms

Definition
Let $A$ be a commutative ring with unity. Let $\struct {B, f}$ and $\struct {C, g}$ be rings under $A$.

Let $h: B \to C$ be a mapping.


 * $(1): \quad h$ is a morphism of rings under $A$.
 * $(2): \quad h$ is a unital algebra homomorphism from the algebra defined by $f$ to the algebra defined by $g$.