= Solution
If $g=xhx^{-1}$ in $G$, applying each <projection map> gives $p_j(g)=p_j(x)p_j(h)p_j(x)^{-1}$ in $G_j$.
Conversely, suppose $p_j(g)$ and $p_j(h)$ are conjugate for every $j$. Define the nonempty finite set
$$
X_j=\{x_j\in G_j:x_jp_j(h)x_j^{-1}=p_j(g)\}.
$$
Every transition map $G_j\to G_i$ carries $X_j$ into $X_i$, so the $X_j$ form an <inverse system>. By the <nonemptiness theorem for inverse limits of finite sets>, there is a compatible tuple $(x_j)\in\varprojlim_jX_j\subseteq G$. Coordinatewise equality then gives $xhx^{-1}=g$. This proves the <finite-quotient criterion for conjugacy in a profinite group>.
Back to article page
