identification vertical bundle and fiber product

Question

Let $\pi:E\rightarrow M$ a vector bundle. I've seen that a vertical vector field on E is a section of $\pi_1:E\times_M E\rightarrow E$, but I thought that it was a section of $\pi|_{VE}:VE\rightarrow E$, where $VE:=\ker T\pi\subset TE$ is the vertical tangent bundle. Then my question is: is there a diffeomorphism between $E\times_M E$ and $VE$ that identify those spaces? Is it the same for vertical 1-form? I found it in Libermann, Symplectic Geometry and Analytical Mechanics. Thank you very much

Answer 1

I guess the isomorphism is the vertical lift \begin{equation} \begin{array}{rcl} Vl_E:E\times_M E&\rightarrow& VE\\ (u_x,v_x)&\mapsto&\frac{d}{dt}|_{t=0}(u_x+tv_x) \end{array} \end{equation}