By the way, here's a counterexample for the most sweeping generalization ("it's always an isomorphism"), which I found online in a book called "Topological Invariants of Stratified Spaces". Let X = [0,1] and F be the skyscraper sheaf Z at 0. Let G be the constant sheaf Z. If U contains 0 then Hom(F|U,G|U)=0, so Hom(F,G)_0 = 0, but of course Hom(Z,Z)=Z.
What about for coherent O_X modules on a ringed space X that need not be a scheme? Say, a complex manifold? Just idle curiosity...
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