Primeness is contraction-closed

Property-theoretic statement
The property of ideals in commutative unital rings of being a prime ideal satisfies the metaproperty of ideals in commutative unital rings of being contraction-closed.

Verbal statement
Given a homomorphism of commutative unital rings, the contraction of a prime ideal in the ring on the right, is a prime ideal in the ring on the left.

Symbolic statement
Suppose $$f:R \to S$$ is a homomorphism of commutative unital rings. Then for any prime ideal $$I$$ of $$S$$, $$I^c = f^{-1}(I)$$ (called the contraction of $$I$$) is a prime ideal of $$R$$.

Importance
This fact allows us to view the spectrum of a commutative unital ring as a contravariant functor, because it allows us to use a homomorphism of commutative unital rings $$f:R \to S$$ to define a backward map $$Spec(f): Spec(S) \to Spec(R)$$, by contraction.