The map to the orbifold base need not be an orbifold map

We give an explicit example of a fibration $f\text{\hspace{0.17em}⁣}:X\to Y$ between smooth projective varieties whose “orbifold base” ${\mathrm{\Delta }}_f$ in the sense of Campana has the property that the induced morphism $X\to (Y,{\mathrm{\Delta }}_f)$ is not a morphism of C-pairs (i.e., it is not an “orbifold morphism”). We however also show that this cannot happen if $f$ is “neat” and $(Y,{\mathrm{\Delta }}_f)$ is sufficiently well-behaved. Finally, we discuss the implications of this statement towards conjectures of Campana aiming to give algebro-geometric characterizations of those varieties which either admit a dense entire curve or a potentially dense set of integral points.