In Dieudonne's Treatise on Analysis (Volume III Section 17.16) the following definition for a linear connection in a vector bundle is given:
A linear connection $C$, in a vector bundle $(E,\pi, M)$ is defined to be a smooth map $C:TM\oplus E\to TE$ which satisfies for each $(k_x,u_x)\in T_xM\oplus E_x$, the following conditions:
- $\pi_E(C(k_x,u_x))=u_x$, where $\pi_E:TE\to E$ is the standard projection
- $T\pi (C(k_x,u_x))=k_x$ where $T\pi:TE\to TM$ is the tangent mapping
- The mapping $k_x\mapsto C(k_x,u_x)$ is a linear mapping of $T_xM$ into $T_{u_x}E$
- The mapping $u_x\mapsto C(k_x,u_x)$ is a linear mapping of $E_x$ into $(TE)_{k_x}$, which is the fiber of $T\pi:TE\to TM$ lying over the base point $k_x\in TM$.
Geometrically this definition makes sense; I interpret $C(k_x,u_x)$ as "the direction I should move within $E$ in order to reach an infinitesimally close fiber, provided that I start at the point $u_x$ of the fiber $E_x$, and move in the direction $k_x$ in the base manifold" (picture below for one-dimensional base and fibers).
With the usual abuse of notation, we have local coordinates $(x,\dot{x},u,\dot{u})\in \Bbb{R}^n\times \Bbb{R}^n\times\Bbb{R}^p\times\Bbb{R}^p$ on $TE$, where $x$ describes the base point, $\dot{x}$ describes the vectors in the fiber tangent to $M$, $u$ describes points in the fibers of $E$ and $\dot{u}$ describes the points tangent to $E$. Relative to such adapted coordinates, the mapping $C$ takes the form $C: (x,\dot{x},u)\mapsto (x,\dot{x},u,-\Gamma_x(\dot{x},u))$, where $\Gamma_x:\Bbb{R}^n\times\Bbb{R}^p\to\Bbb{R}^p$ is bilinear (the minus sign is merely convention to make things match up with the usual Christoffel symbols).
Question.
Suppose we are given two vector bundles $E,F$ over $M$, with respective linear connections $C_E,C_F$. How can I define abstractly (i.e without any references to coordinates) a connection $C_{\text{Hom}(E,F)}$ in the bundle $\text{Hom}(E,F)$? I have already managed to define it in coordinates, meaning I figured out how, given local coordinates, to define the $\Gamma$ for $\text{Hom}(E,F)$ in terms of $\Gamma$ for $E$ and $F$. Then, I used the appropriate transformation laws for $\Gamma$'s to check that everything works out so that we do indeed obtain a globally well-defined mapping $\text{C}_{\text{Hom}(E,F)}$.
I will include this local definition if anyone requests it; but the notation is a little heavy so I suppress it for now. The reason I ask this question is because defining things locally and then verifying the compatibility conditions is a little tedious, especially since $\Gamma$ has a somewhat complicated transformation behavior.
By the way, I have already determined how to define connections on $E\oplus F$ and $E\otimes F$, in an abstract manner (and also locally). Also, I'm aware that $\text{Hom}(E,F)\cong E^*\otimes F$, so if someone can provide an abstract definition for $C_{E^*}$ in terms of $C_E$ that is also sufficient. Thank you.
Picture: