  We prove that for any nondegenerate dendrite $D$ there exist topologically
mixing maps $F : D \to D$ and $f : [0, 1] \to [0, 1]$, such that the natural
extensions (aka shift homeomorphisms) $\sigma_F$ and $\sigma_f$ are conjugate,
and consequently the corresponding inverse limits are homeomorphic. Moreover,
the map $f$ does not depend on the dendrite $D$, and can be selected so that
the inverse limit $\underleftarrow{\lim} (D, F)$ is homeomorphic to the
pseudo-arc. The result extends to any finite number of dendrites. Our work is
motivated by, but independent of, the recent result of the first and third
author on conjugation of Lozi and H\'enon maps to natural extensions of
dendrite maps.
