 Finite-quotient criterion for conjugacy in a profinite group

Elements $g,h$ of a profinite group $G={\underleftarrow{\lim }}_j{G}_j$ are conjugate exactly when their images are conjugate in every $G_j$. For the nontrivial direction, the finite nonempty sets of conjugators in the $G_j$ form an inverse system, whose inverse limit supplies a conjugator in $G$.