I want to prove that $0 \to F\to G\to H \to 0$ is an exact sequence of étale sheaves. I understand that it is enough to show that $0\to F_{\bar{x}}\to G_{\bar{x}}\to H_{\bar{x}}\to 0$ is exact at every geometric point $\bar{x}\to X$. Why is it then enough to show that $0\to F(U)\to G(U)\to H(U)\to 0$ is exact for every strictly Hensel local scheme $U$?

I ask this because I don't understand the proof of proposition 6.12 of Voevodsky's "Lecture notes on motivic cohomology". Specifically Im trying to understand the first two paragraphs here http://www2.maths.ox.ac.uk/cmi/library/MotivicCohomologyold.pdf#page=52 and use that argument to understand the proof of theorem 7.20 ( Suslin's rigidity theorem ) here http://www2.maths.ox.ac.uk/cmi/library/MotivicCohomologyold.pdf#page=63. It seems the arguments are quite similar in that the book states that it is enough to show an isomorphism on stalks and then goes to show said isomorphism on the sheaf valuated on some henselian schemes. Is this correct?