  We consider the one and the two obstacles problems for the nonlocal nonlinear
anisotropic $g$-Laplacian $\mathcal{L}_g^s$, with $0<s<1$. We prove the strict
T-monotonicity of $\mathcal{L}_g^s$ and we obtain the Lewy-Stampacchia
inequalities. We consider the approximation of the solutions through semilinear
problems, for which we prove a global $L^\infty$-estimate, and we extend the
local H\"older regularity to the solutions of the obstacle problems in the case
of the fractional $p(x,y)$-Laplacian operator. We make further remarks on a few
elementary properties of related capacities in the fractional generalised
Orlicz framework, with a special reference to the Hilbertian nonlinear case in
fractional Sobolev spaces.
