Let $B$ a topologiocal space (I'm not sure if it should be nice enough for the following statement ... eg paracompact).
Let $L$ be a line bundle over $B$.
My question is how to see that the morphism bundle $\underline{Hom}(L,L)$ is a trivial bundle? (here $\underline{Hom}(L,L)$ means that it is fiberwise $b$ the space linear maps between one dimensional spaces $L_b$)
My attempts: Obvioulsy it suffice to show that there exist a non vanishing glocal section $B \to \underline{Hom}(L,L)$.
Intuitively I would guess that the map $b \to id_{L_b}\in Hom(L_b,L_b)$ should work. Is it ok or is it a bit more complicated and I have overseen a subtle detail?