Let $G$ be a profinite group and all its finite quotients be $G_i$. Let $\phi_i : G \longrightarrow G_i$ be standard map and $\phi_{i,j}: G_j \longrightarrow G_i$ be the standard surjective maps between $G_i$'s (part of inverse system). And let $\psi_i : G \longrightarrow G_i$ be a collection of map such that $\phi_{i,j} \circ \psi_{j} = \psi_i$ for every pair $i, j$, where $\phi_{i,j}$ makes sense. Then how can we lift $\psi_i$ to $\psi: G \longrightarrow G$ such that $\psi \circ \phi_i = \psi_i$? Is it obvious that such $\psi$ will be continuous endomorphism of $G$?
Thanks!