Correspondence theorem between rings problem

I am looking at the correspondence theorem for rings. So if $I$ is a 2 sided ideal in $R$ and $\psi $ is the natural homomorphism from $R$ to $R/I$ then there is a bijection between the set of two sided ideals in $R$ that contain $I$ and the set of 2 sided ideals in $R/I$, namely $\phi (J)= \psi ^{-1} (J) $.

My question is, the proof involves showing that if $K$ is a two sided ideal in $R$ containing $I$ then $\psi (K) $ is a two sided ideal in $R/I$. When proving this part I fail to see how the condition that $I $ is contained in $K$ is necessary. My proof for this part seems to work even if $K$ does not contain $I$ but this can’t be true because $\phi $ is a bijection.