The Stacks project

Proof. Given a deformation of $P$ we can take the strict henselization of it to get a deformation of the strict henselization; this is clear and we encourage the reader to skip the proof. More precisely, let $(A, Q) \to (k, P)$ be a morphism in $\mathcal{F}$, i.e., an object of $\mathcal{D}\! \mathit{ef}_ P$. Since the kernel of the surjection $Q \to P$ is nilpotent, we find that $Q$ is a local ring with the same residue field as $P$. Let $Q^{sh}$ be the strict henselization of $Q$. Recall that $Q \to Q^{sh}$ is flat (More on Algebra, Lemma 15.45.1) and hence $Q^{sh}$ is flat over $A$. By Algebra, Lemma 10.156.4 we see that the map $Q^{sh} \to P^{sh}$ induces an isomorphism $Q^{sh} \otimes _ A k = Q^{sh} \otimes _ Q P = P^{sh}$. Hence $(A, Q^{sh}) \to (k, P^{sh})$ is the desired object of $\mathcal{D}\! \mathit{ef}_{P^{sh}}$. $\square$