Recently I posted a question about the horizontal/vertical decomposition of the double tangent space. In it, I invoked the below theorem. I cannot seem to find a good proof of it, specifically of the part where the vectors $X$ and $Y$ are defined.
To be sure, I have found a seemingly good one in the below reference; however, the book is in German and my attempts to translate it have been faulty. Page 43 1975 edition:
D. Gromoll, W. Klingenberg, W. Meyer. Riemannsche geometric im groben
Theorem. Let $M$ be a differential manifold with an affine connection $\nabla$. Then, there exists a unique differentiable function $K: TTM \longrightarrow TM$, called the connection function, that satisfies the following conditions:
(1) If $v \in TM$ and $\pi(v) = p$, where $\pi: TM \longrightarrow M$ is the canonical projection, then $K(T_v TM) \subseteq T_p M$.
(2) $K_v := K|_{T_v TM}: T_v TM \longrightarrow T_p M$ is a linear map.
(3) If $v \in TM$ and $w \in T_v TM$ are such that $w = d_p X(Y)$, where $Y \in T_p M$ and $X \in {X \in \mathfrak{X}(M): X(p) = v}$, then $K(w) = \nabla_Y X$.
Let $E$ be a vector bundle over $M$ and denote the natural projection map by \begin{align*} \pi_E: E &\rightarrow M. \end{align*} Given $p \in M$ and $e \in E_p$, observe that the pushforward map of $\pi_E$ defines a linear map $$ (\pi_E)_*: T_eE \rightarrow T_pM $$
Given an affine connection $\nabla$ on $E$, we want to show that there is a natural bundle map $$ K: TE \rightarrow E $$ such that if $s$ is a section of $E$ on a neighborhood of $p\in E$ and $X\in T_pM$, then $$ \nabla_Xs(p) = K(s_*(X)).$$ Given any $p \in M$, $e \in E_p$ and $\dot{e} \in T_eE$, there is a curve $s: I \rightarrow E$ such that $s(0) = e$ and $s'(0) =\dot{e}$. Then $s$ is a section of $E$ along the curve $c = \pi_E\circ s: I \rightarrow M$. Taking the covariant derivative of $s$, let $$ K_e(\dot{e}) = \nabla_Xs(0) \in E_p, $$ where $X=(\pi_E)_*(s’(0))$. If you write this all out with respect to a frame of sections of $E$, you can confirm this is a bundle map.