about the subbundle in complex geometry by daniel huybrechts

Question

This question coming from page 68 of the book Compex Geometry by Daniel Huybrechts. Here i write the question: we have two holomorphic vector bundle $E,F$ given by the cocycles $\psi_{ij}$ and $\psi^{'}_{ij}$ respectively.
The exercise says that if for all $x \in U_i\cap U_j$ the matrix of $\{ \psi^{'}_{ij} \}$ has the form $\begin{bmatrix} \psi_{ij}&* \\ 0& \phi_{ij} \end{bmatrix}$ than $E$ is a holomorphic sub-bundle of $F$, i.e. there exist a canonical injection $E \subset F$.
Conversely if $E$ is a sub-bundle of $F$ we can find cocycles of this form and the cokernel $F/E$ is described by the cocycles $\phi_{ij}$.
Can someone helping to show this fact? I can't figurt out why the matrix is not a diagonal matrix?