Let $E\rightarrow M$ a holomorphic vector bundle over a complex manifold M, and $S\subset E$ a holomorphic subbundle. Let $\tilde{\eta}$ a local holomorphic section on the quotient bundle $Q=E/S$ (defined over a open subset $U\subset M$). My question is how can i ensure there exists a local holomorphic section $\eta$ on $E$ representing $\tilde\eta$?
This is part of a proof in Kobayashi's Differential geometry of complex vector bundles, proposition I.6.6. My first idea was to take a trivialization function on $U$ in both bundles $E$ and $Q$ and look for a relation between the local representation of $\tilde\eta$ and this trivialization functions, but i can't figure it out how can i obtain $\eta$.