Canonical choice of inverse system for profinite set.

Question

Let $X$ be a profinite set - an inverse limit $\varprojlim X_i$. How can one prove that then $X=\varprojlim Y_i$, where $Y_i$ is finite quotient spaces of $X$? I may prove it if $X$ is topological group, since $\varprojlim$ is left-exact, $X=\varprojlim X/K_i=\varprojlim Y_i$ where $K_i$ is kernel of $X \to X_i$. But there aren't kernels in topological case.