I'm looking for a reference for the following statement:


Let $P$ be a property of morphisms of schemes local on the target in the etale topology. Let $f : X\rightarrow Y$ be a morphism of schemes which is locally of finite presentation, and such that for all points $y\in Y$, the restriction of $f$ to the strict henselian local ring $\mathcal{O}\_{Y,y}^{sh}$ has property $P$. Then $f$ has property $P$.


(Is the LoFP condition necessary?)


This seems to follow from various results in the stacks project, but I can't find it explicitly stated anywhere except for the specific case where P = "flat" (tag 05VL).


I need this result for an appendix in my thesis. Currently I've stitched a proof together using various results of the stacks project, but it seems more elegant to cite a reference that proves this explicitly if possible.



etale localization reference request
That is not true without further hypotheses, but it is true with one additional hypothesis. First, here is a counterexample. Let $P$ be the property that the morphism $f$ is quasi-compact. This property is local for the étale topology, and even for the fpqc topology, cf. <http://stacks.math.columbia.edu/tag/02KQ> for instance. 


Let $Y$ be $\text{Spec}\ \mathbb{Z}$, and let $X$ be the disjoint union over all primes $p$ of $\text{Spec}\ \mathbb{Z}/p\mathbb{Z}$. There is a unique morphism $f:X\to Y$. This morphism is not quasi-compact since $X$ is not quasi-compact. The morphism $f$ is locally of finite presentation since every point of $X$ has an open neighborhood that is isomorphic to $\text{Spec}\ \mathbb{Z}/p\mathbb{Z}$ for some prime $p$.


On the other hand, for every closed point $y=\langle p\rangle$ of $Y$ the base change of $f$ over the local ring $\mathcal{O}\_{Y,y} = \mathbb{Z}\_{\langle p \rangle}$ is the quasi-compact, even affine, morphism corresponding to the ring homomorphism $\mathbb{Z}\_{\langle p \rangle} \to \mathbb{Z}\_{\langle p \rangle}/p\mathbb{Z}\_{\langle p \rangle}$. Thus the base change of $f$ to every strict Henselization is also quasi-compact.


The simplest additional hypothesis guaranteeing your result is "compatible with filtered limits of schemes / filtered colimits of rings": for every affine scheme $Y=\text{Spec}\ A$ that is a filtered colimit $A = \varinjlim A\_\lambda$ of rings $(A\_\lambda)\_{\lambda\in I}$, for every compatible family of morphisms $$f\_\lambda:X\_\lambda \to \text{Spec}\ A\_\lambda, \ \ \phi\_{\mu,\lambda}:X\_\mu \xrightarrow{\cong} X\_{\lambda}\otimes\_{A\_\lambda} A\_\mu,$$ with filtered limit $f:X\to \text{Spec}\ A$, then $f$ has property $P$ if and only if there exists some $\lambda$ in $I$ such that for all $\mu>\lambda$, $f\_\mu$ has property $P$. There are many examples of such properties in EGA $\textrm{IV}\_3$, Section 8.10, pp. 36-41. (Please note, in EGA, there is a standing hypothesis that every $f\_\lambda$ is finitely presented, i.e., locally finitely presented and quasi-compact. Also I have stated "compatible with limits" slightly incorrectly because I am only considering the case that $Y$ is affine. The correct formulation is in EGA. If your property is étale local on the base, then you can always reduce to the affine case.) 


Since the strict henselization $A$ of a Noetherian ring $B$ at a prime $\mathfrak{p}$ is a filtered colimit of finitely presented, étale $B$-algebras, if your property is compatible with limits and is also étale local, then it holds on $\text{Spec} B$ if and only if it holds after base change to each strict Henselization $\text{Spec}\ B\_{\mathfrak{p}}^{sh}$.


