Let $ \pi: \mathcal{E}= M \times E \rightarrow M $ be a trivial vector bundle (where $M$ is smooth and $E$ is a finite dimensional real vector space).
Let $\nabla = d + \omega $ be a covariant derivative on $\mathcal{E}$, where $\omega \in \mathcal{A}^1(M,End(\mathcal{E})).$ Let $\nabla^{\pi^* \mathcal{E}}$ denotes the induced covariant derivative on the bundle $\pi^* \mathcal{E} \rightarrow \mathcal{E}$
Question : Let $s \in \Gamma(\mathcal{E},\pi^* \mathcal{E})$ be the smooth section which to a point $v \in \mathcal{E}$ assigns the point $(v,v) \in \pi^* \mathcal{E}$, and Let $(X,V)$ be a vector field on $\mathcal{E}$. What is $\nabla^{\pi^* \mathcal{E}}_{(X,V)}s ?$
My attempt: Let $s' \in \Gamma(M, \mathcal{E})$ be the section such that $s= \pi^*s'$, then applying the definition of the induced covariant derivative on the pullback bundle we get $$ \nabla^{\pi^* \mathcal{E}}_{(X,V)}s = \nabla_{\pi_*(X,V)}s' = (ds')(\pi_*(X,V))+ \omega(\pi_*(X,V))s' = ds'(X)+ \omega(X)s' . $$ However in the proof of proposition 1.20 in the book Heat kernels and Dirac operators page 28 the answer was $$\nabla^{\pi^* \mathcal{E}}_{(X,V)}s = V +\omega(X)v,$$ Where $v=s(x,v)$. Where is my mistake?
The issue here is that $s$ is not the pullback of any section of $\mathcal{E}\to M$. The space of sections of $\pi^*\mathcal{E}$ is however generated by these pullbacks.
Fix a basis $e_1,\dots,e_k$ of $E$, and for each $a=1,\dots,k$ let $s_a$ be the section of $\mathcal{E}$ defined by $$s_a(x):=(x,e_a).$$ Then by the definition of $s$ we have for every $x\in M$ and $v\in E$ $$s(x,v)=((x,v),v)=\left((x,v),\sum_{a=1}^kv^ae_a\right)=\sum_{a=1}^kv^a(\pi^*s_a)(x,v).$$ From now on, I will use the Einstein summation on repeated indices. Let $f^a:E\to\mathbb{R}$ be the function assigning to each $v\in E$ the $a$-th component of $v$ in the basis $(e_1,\dots,e_k)$. From the definition of the pullback connection, we have \begin{equation*} \begin{gathered} \left(\nabla^{\pi^*\mathcal{E}}_{(X,V)}s\right)(x,v)=\left(\mathrm{d}_{(X,V)}f^a\right)(v)\pi^*(s_a(x))+f^a(v)\pi^*\left(\nabla_Xs_a\right)=\\ =V^a\pi^*(s_a(x))+f^a(v)\pi^*\left(\omega_x(X)s_a(x)\right)=V+\omega(X)v \end{gathered} \end{equation*} using that $s_a$ are constant sections and that $f^a(v)s_a=v$.