I am reading the book Complex Geometry: an Introduction by Daniel Huybrechts. On page 180 the author defined the Atiyah class as follows:
The Atiyah class $$A(E)\in H^1(X,\Omega_X\otimes \mathrm{End}(E))$$ of the holomorphic vector bundle E is given by the $\mathrm{\check{C}}$ech cocycle $$A(E)=\{U_{ij},\psi_j^{-1}\circ(\psi_{ij}^{-1}d\psi_{ij})\circ\psi_j\}.$$
Here $X$ is a complex manifold and $X=\cup U_i$ is an open covering such that $\psi_i\colon E|_{U_i}\cong U_i\times\mathbb{C}^r$ are holomorphic trivializations, and $\Omega_X$ is the sheaf of holomorphic 1-forms on $X$. Moreover, $U_{ij}:=U_i\cap U_j.$
I cannot see why these $\psi_j^{-1}\circ(\psi_{ij}^{-1}d\psi_{ij})\circ\psi_j$ are in sections $\Gamma(U_{ij}, \Omega_X\otimes \mathrm{End}(E)).$ If this is true, take some $x\in U_{ij},$ $v\in T_xX$ and $e\in\Gamma(U_{ij},E),$ then $v\otimes e(x)$ should be mapped by $\psi_j^{-1}\circ(\psi_{ij}^{-1}d\psi_{ij})\circ\psi_j(x)$ to an element in $\mathbb{C}\otimes E.$ But I think the result should still be in $T_xX\otimes E,$ as $d\psi_{ij}$ is from $T(U_{ij}\times\mathbb{C}^r)$ to $T(U_{ij}\times\mathbb{C}^r)$. Where am I wrong? Could someone help me?