Lemma 09XI in the stacks project includes a characterization (#5) by Gabber of Henselian pairs $(A,I)$: first, $I\leq \mathrm J(A)$ is contained in the Jacobson radical and second, every monic $f\in A[x]$ which is residually of the form $x^{\deg f-1}(x-1)\in \frac{A}{I}[x]$ has a root in $1+I$.
The proof uses this condition #5 to prove an $A$-algebra morphism from an étale $A$-algebra to $A/I$ lifts to a retraction of the étale $A$-algebra. The actual source for a polynomial as above is lemma 0EM0, whose statement and proof are fairly involved and opaque to me.
Question. Let $A$ be a commutative ring and $I\leq \mathrm J(A)$ an ideal contained in the Jacobson radical. Where do monics $f\in A[x]$ which are residually of the form $x^{\deg f-1}(x-1)\in \frac{A}{I}[x]$ "come from" conceptually? Algebraic and geometric answers are both welcome.
