An unclear sentence in the book "Griffiths, Harris-Principles of algebraic geometry"

Solution 1:

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*}