 Tensoring with an invertible sheaf is exact

For an invertible sheaf $L$, tensoring sheaves of modules with $L$ is an exact functor: on an open cover trivializing $L$, it is naturally the identity functor. Equivalently, its stalks are free modules of rank one and hence flat modules. Its inverse functor is tensoring with the dual of a sheaf $L^{\vee }$, because $L\otimes L^{\vee }\cong {\mathcal{O}}_X$. This justifies untwisting a surjection onto a positively twisted coherent sheaf.