ag.algebraic geometry – A question on effective divisors

Let $X$ be a projective variety with two morphisms $f:Xrightarrow Y$ and $g:Xrightarrow Z$. Assume that $Pic(X) = f^{*}Pic(Y)oplus g^{*}Pic(Z)$. Then if $D$ is a divisor on $X$ we can write $D = f^{*}D_Y + g^{*}D_Z$, where $D_Y,D_Z$ are divisors on $Y$ and $Z$ respectively.

If $D$ is effective are then $D_Y$ and $D_Z$ effective as well?

This holds for instance when $X = mathbb{P}^n times mathbb{P}^m$ is a product and $f,g$ are the projections onto the factors.