I have the following situation: $E,F$ are (smooth) vector bundles over a smooth manifold $M$. Assume we are given connections $\nabla^E,\nabla^F$ on $E,F$ and a homomorphism of $M$-bundles $P:E\rightarrow F$.
Note that $\nabla^E$ induces a connection on the dual bundle $E^*$, which we denote by $\nabla^{E^*}$.
$\nabla^{E^*}, \nabla^{F}$ induce a connection on the tensor product bundle $E^*\otimes F \cong Hom(E,F)$, which we denote by $\nabla^{E^*\otimes F}$.
Here is the question:
Let $X\in \Gamma(E)$ . The homomorphism $P$ can be identifed with a $C^\infty(M)$-module homomorphism: $\hat P:\Gamma(E)\rightarrow \Gamma(F)$. We can also identify $P$ as a section of the Hom-bundle: $\bar P\in \Gamma(Hom(E,F))=\Gamma(E^*\otimes F)$.
Now take $V\in \Gamma(TM),X\in \Gamma(E)$. Is the following equality true?
$(\nabla_V^{E^*\otimes F} \bar P)(X)=\nabla_V^F (\hat P(X)) \in \Gamma(F)$
In other words, I am asking whether we can change the order of differentiation and action on sections.
At first glance there is something strange here, since the right side is independent of the connection on $E$.
In more detail:
1) $(\nabla_V^{E^*\otimes F} \bar P)(X). $
Here I think of ($\nabla_V^{E^*\otimes F} \bar P)$ as a section of the hom-bundle, and using the standard identification I can think of it as a bundle-homomorphism, and then (another identification) as a homomrphims of (modules of) sections $\Gamma(E)\rightarrow \Gamma(F)$.
2) $\nabla_V^F (\hat P(X)).$
Here I first use $\hat P$ to get a section $F$ and then I differentiate it covariantly along $V$.