Comparison of component groups of -adic and mod monodromy groups

Let \ρ_:GalKₙ(Q_)_ be a semisimple compatible system of -adic representations of a number field K that is arising from geometry. Let G__n,Q_ and G_n,F_ be respectively the algebraic monodromy group and full algebraic envelope of ρ_. We prove that there is a natural isomorphism between the component groups π₀(G_) π₀(G_) for all sufficiently large .