  f-Harmonic maps were first introduced and studied by Lichnerowicz in
\cite{Li} (see also Section 10.20 in Eells-Lemaire's report \cite{EL}). In this
paper, we study a subclass of f-harmonic maps called f-harmonic morphisms which
pull back local harmonic functions to local f-harmonic functions. We prove that
a map between Riemannian manifolds is an f-harmonic morphism if and only if it
is a horizontally weakly conformal f-harmonic map. This generalizes the
well-known Fuglede-Ishihara characterization for harmonic morphisms. Some
properties and many examples as well as some non-existence of f-harmonic
morphisms are given. We also study the f-harmonicity of conformal immersions.
