In Huybrechts"Complex Geometry"P93,he wrote:"a sequence of holomorphic vector bundles $0\rightarrow E\rightarrow F\rightarrow G\rightarrow 0$ is exact if and only if $0\rightarrow G^*\rightarrow F^*\rightarrow E^* \rightarrow 0$ is exact.
As we know,for all $x \in U_i\cap U_j$ the matrix of $\{ \psi^{'}_{ij} \}$ has the form $\begin{bmatrix} \psi_{ij}&* \\ 0& \phi_{ij} \end{bmatrix}$ ,then $E$ 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 $\phi_{ij}$.
S0,For $F^*$,the matrix of $(\{ \psi^{'}_{ij} \}^{-1})^{T}$ has the form $\begin{bmatrix} (\{ \psi_{ij} \}^{-1})^{T}&0 \\ *^{'}& (\{ \phi_{ij} \}^{-1})^{T}\end{bmatrix}$ .
I wonder that why the claim is not "$0\rightarrow E^*\rightarrow F^*\rightarrow G^* \rightarrow 0$ is exact"?
Note that a morphism of vector bundle is uniquely determined by a collection of holomorphic maps $$\lbrace f_i: U_i \longrightarrow\mathcal{M}_{m\times n}(\mathbb{C}) \rbrace_i $$ such that $f_i=(\{\psi^{'}_{ij} \}^{-1})^{T}f_j(\{ \psi_{ji} \}^{-1})^{T}$. At this point I can define $$f_i=\left[\begin{array}{ccc} \mathbb{I}_m\\ O \\ \end{array} \right].$$
I prefer a detailed explanation,thanks a lot!