The Stacks project

At the start of the proof, the wrong lemma is cited: should cite Lemma 09XK. In the second sentence, the invocation of the universal property doesn't make any sense unless $J$ lands in $I^h(A^h \otimes_A B) = I(A^h \otimes_A B)$, which would hold if $J=IB$. But there is the flexibility to replace $J$ with any bigger ideal of $B$ without affecting the henselian hypothesis on $(B, J)$, so you should really assume $V(J)=V(IB)$: this would ensure $(B,IB)$ is also henselian by Lemma 09XJ and would ensure that when $A$ is local with maximal ideal $I$ we can take $J$ to be the Jacobson radical of $B$ (rather than demanding $J=IB$, which would be an unpleasant requirement for such cases).

I should have also mentioned that currently this Lemma is only mentioned in two proofs, neither of which is impacted by imposing the requirement $V(J)=V(IB)$ (though the second proof which claims to invoke this Lemma doesn't really clearly indicate where it is used -- that proof says this Lemma is used, but never explicitly invokes it with a cross-reference at any step).

Thank very much indeed. Fixed here.

Suggested slogan: "Henselization commutes with base change along integral maps"