Why are ideals of a ring this way?

Question

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I understand the proofs. But why do we need the last condition to prove correspondence theorem? IE: why do we have to show $R/I$ is isomorphic to $R’/I’$? Or is it just a side fact that follows? I think that just proving the first $3$ facts is enough to show the objection exists and the fourth one is just a side consequence. Is this true? Why

This is from Artin Abstract Algebra

Answer 1

Two paragraphs above Proof of the Correspondence Theorem:

If the ideal $I$ of $R$ corresponds to the ideal $\mathcal{I}$ of $\mathcal{R},$ the quotient rings $R/I$ and $\mathcal{R}/\mathcal{I}$ are naturally isomorphic.