The Stacks project

Proof. Let $E \subset X'$ be the exceptional divisor of $b$ which is an effective Cartier divisor by Lemma 31.32.4. Then $(b')^{-1}E$ is an effective Cartier divisor on $X''$ by Lemma 31.32.11. Let $E' \subset X''$ be the exceptional divisor of $b'$ (also an effective Cartier divisor). Consider the effective Cartier divisor $E'' = E' + (b')^{-1}E$. By construction the ideal of $E''$ is $(b \circ b')^{-1}\mathcal{I} (b \circ b')^{-1}\mathcal{J} \mathcal{O}_{X''}$. Hence according to Lemma 31.32.5 there is a canonical morphism from $X''$ to the blowup $c : Y \to X$ of $X$ in $\mathcal{I}\mathcal{J}$. Conversely, as $\mathcal{I}\mathcal{J}$ pulls back to an invertible ideal we see that $c^{-1}\mathcal{I}\mathcal{O}_ Y$ defines an effective Cartier divisor, see Lemma 31.13.9. Thus a morphism $c' : Y \to X'$ over $X$ by Lemma 31.32.5. Then $(c')^{-1}b^{-1}\mathcal{J}\mathcal{O}_ Y = c^{-1}\mathcal{J}\mathcal{O}_ Y$ which also defines an effective Cartier divisor. Thus a morphism $c'' : Y \to X''$ over $X'$. We omit the verification that this morphism is inverse to the morphism $X'' \to Y$ constructed earlier. $\square$