  Let
$\{\rho_{\ell}:\mathrm{Gal}_K\to\mathrm{GL}_n(\mathbb{Q}_{\ell})\}_{\ell}$ be a
semisimple compatible system of $\ell$-adic representations of a number field
$K$ that is arising from geometry. Let
$\textbf{G}_{\ell}\subset\mathrm{GL}_{n,\mathbb{Q}_{\ell}}$ and
$\widehat{\underline{G_{\ell}}}\subset\mathrm{GL}_{n,\mathbb{F}_\ell}$ be
respectively the algebraic monodromy group and full algebraic envelope of
$\rho_{\ell}$. We prove that there is a natural isomorphism between the
component groups $\pi_0(\textbf{G}_{\ell}) \simeq
\pi_0(\widehat{\underline{G_\ell}})$ for all sufficiently large $\ell$.
