Let $E,F$ be two holomorphic vector bundles on a compact Kahler manifold $X$. Denote by $\mathbb{P}(E), \mathbb{P}(F)$ the associated projective bundles and $L_E=\mathcal{O}_E(-1), L_F=\mathcal{O}_F(-1)$ the tautological line bundles. Let $f:E\to F$ a bundle map.
Is there a "canonical" induced map $$\tilde{f}:\tilde{L}_E\to \tilde{L}_F$$ where $\tilde{L}_E,\tilde{L}_F\to Y$ are some pull back of $L_E,L_F$ to some space $Y$?