We prove that if
f
:
X
→
Z
f\colon X\to Z
is a smooth surjective morphism between projective manifolds and if
−
K
X
-K_X
is semi-ample, then
−
K
Z
-K_Z
is also semi-ample. This was conjectured by Fujino and Gongyo. We list several counterexamples to show that this fails without the smoothness assumption on
f
f
.
We prove the above result by proving some results concerning the moduli divisor of the canonical bundle formula associated to a klt-trivial fibration
(
X
,
B
)
→
Z
(X,B)\to Z
.