  We study a geometrical condition (PHWC) which is weaker than horizontal weak
conformality. In particular, we show that harmonic maps satisfying this
condition, which will be called {\em pseudoharmonic morphisms}, include
harmonic morphisms and can be described as pulling back certain germs to
certain other germs. Finally, we construct a canonical f-structure associated
to every map satisfying (PHWC) and find conditions on this f-structure to
ensure the harmonicity of the map.
