Answer by Captain Lama for Correspondence theorem between rings problem

Let us consider three sets: $A$ is the set of ideals of $R$, $B\subset A$ is the set of ideals of $R$ which contain $I$, and $C$ is the set of ideals of $R/I$.

Then as you noticed, $K\mapsto \psi(K)$ defines a map $A\to C$. It is not a bijection, but only a surjection, and the correspondence theorem says that its restriction to $B\subset A$ does define a bijection $B\to C$, with inverse $J\mapsto \psi^{-1}(J)$.

Sanity check: $\psi(\psi^{-1}(J))=J$ for all $J\in C$, but for $K\in A$, $\psi^{-1}(\psi(K))$ is equal to $K$ if and only if $K\in B$. All good.