I am trying to understand the proof of Theorem 3.7, page 72 of Riemannian Geometry by M. Do Carmo.
For $M$ a Riemannian manifold and $(U,\varphi)$ a chart around a point $p\in M$, he (more or less) defines a map $$F: TU\to M\times M$$ by $F(q,v)=(q,\exp_q v)$. He then asserts that the matrix of $dF_{(p,0)}$ in coordinates $(TU, T\varphi)$ and $(U\times U,\varphi\times\varphi)$ is $$\left(\begin{matrix} I&I\\ 0&I \end{matrix}\right),$$ because $(d\exp_p)_0=I$.
I really do not understand why! For me, the expression of $F$ in coordinates is $$(x_1,\dots,x_n,v_1,\dots,v_n)\mapsto (x_1,\dots,x_n,\exp_{x_1,\dots,x_n}(v_1,\dots,v_n))...$$
Write $F(p,v)=(G_1(p,v),G_2(p,v))$, where $G_1(p,v)=p$ and $G_2(p,v)=exp_p(v)$. Note that $$d_p G_1(p,0)=I\ \mbox{and} \ d_vG_1(p,0)=0 $$
On the other hand $$d_p exp_p(0)=I\ \mbox{and}\ d_v exp_p(0)=I$$