Let $X$ be a complex manifold and consider a holomorphic vector bundle $\pi:E\to X$ of rank $r$. By the usual correspondence between vector bundles and locally free sheaves, we consider $E$ as a sheaf by abuse of notation. We denote with $\mathscr A_X^{p,q}$ the sheaf of $C^\infty$ $(p,q)$-differential forms on $X$ and we define: $$A^{p,q}(E):=\Gamma(X,\mathscr A^{p,q}\otimes_{\mathscr O_X} E)$$ $$A^k(E):=\bigoplus_{p+q=k}A^{p,q}(E)$$
Now suppose that $\nabla:A^\bullet(E)\to A^\bullet(E)$ is a linear connection on $E$, then the curvature operator is defined as: $$\nabla^2:A^0(E)\to A^2(E)$$ where $\nabla^2$ is the composition of the maps $\nabla: A^0(E)\to A^2(E)$ and $\nabla: A^1(E)\to A^2(E)$. One can show that $\nabla^2$ is $C^\infty(X)$-linear.
I don't understand the following sentence from the Griffiths-Harris book at page 75 (I rephrase it by using my notation):
Since $\nabla^2$ is $C^\infty(X)$-linear, then it is induced by a bundle map $E\to\bigwedge^2T^*X\otimes E$, or in other words, $\nabla^2$ corresponds to a global section $\Theta$ of the bundle $\bigwedge^2T^*X\otimes\operatorname{Hom}(E,E)$.
I absolutely don't understand how I can see the operator $\nabla^2$ as the section $\Theta$.
Let $E$, $F$ be two smooth vector bundles over a smooth manifold $X$. A linear map $L : \Gamma(E) \to \Gamma(F)$ is $C^{\infty}(X)$-linear if and only if there is a vector bundle homomorphism $\sigma : E \to F$, i.e. $\sigma \in \Gamma (\operatorname{Hom}(E, F))$, such that $L(s) = \sigma\circ s$ for all $s \in \Gamma(E)$. A proof can be found in Lee's Introduction to Smooth Manifolds (second edition), Lemma 10.29 where is goes by the name 'Bundle Homomorphism Characterization Lemma'.
In this case $F = \bigwedge^2T^*X\otimes E$. To complete the identification, note that $\operatorname{Hom}(E, F) \cong E^*\otimes F$ and hence
\begin{align*} \operatorname{Hom}\left(E, \bigwedge\nolimits^2T^*X\otimes E\right) &\cong E^*\otimes \bigwedge\nolimits^2T^*X\otimes E\\ &\cong \bigwedge\nolimits^2T^*X\otimes E^*\otimes E\\ &\cong \bigwedge\nolimits^2T^*X\otimes \operatorname{Hom}(E, E). \end{align*}