Effect of ideal contraction on Galois correspondent

This fact is an application of the following pivotal fact/result/idea: nilradical of subring lemma
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Statement

Suppose $f:R\to S$ is a homomorphism of commutative unital rings, and $I$ is an ideal of $S$. Suppose $\mathcal{Z}(I)$ denotes the subset of $Spec(S)$ comprising the prime ideals which contain $I$ (the Galois correspondent to $I$ under the Galois correspondence between a ring and its spectrum). Then:

Failed to parse (syntax error): {\displaystyle \mathcal{Z}(I^c) = \overline{f^*\mathcal{Z}(I))}