Let $\pi: E \longrightarrow M$ be a vector bundle, $e \in E$ and $h: T_{\pi(e)}M \longrightarrow T_eE$, $X \mapsto \tilde{X}=\sigma_*(X)$, where $\sigma$ is any local section of $E$ which is parallel in the direction of $X$ at $p$ and satisfies $\sigma(p)=e$.
In "Lectures on Kähler Geometry" by Andrei Moroianu the horizontal subspace $T^h_{e}E$ is defined as $\text{im}(h)$. Furthermore it is claimed that $\pi_* \circ h = Id$. Can you help me understand, why this is true?
In a vector bundle $\pi \circ \sigma = Id_M$ for all $\sigma \in \Gamma(E)$.
$(\pi_* \circ h)(X) = \pi_*(h(X)) = \pi_*(\sigma_*(X)) = (\pi_* \circ \sigma_*)(X) = (\pi \circ \sigma)_*(X) = (Id_M)_*(X) = X$