Equivalence of Definitions of Unital Associative Commutative Algebra/Homomorphisms

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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.

The following are equivalent:

$(1): \quad h$ 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 $g$.