Connection giving an identification of the double-tangent bundle

Question

Let $M$ be a manifold, and let $\nabla$ be a (possibly torsion-free) connection on $M$. Then I am pretty sure that $\nabla$ induces an isomorphism between the double-tangent bundle and the Whitney sum of two copies of the tangent bundle $$T_{(m,\hat{m})}TM\cong T_mM\oplus T_mM$$ as follows: let an element of $T_{(m,\hat{m})}TM$ be given as the equivalence class of a path $(\gamma(t),v(t))$ in $TM$, i.e. $v(t)\in T_{\gamma(t)}M$. Then we identify $$[\gamma(t),v(t)] = \left(\left.\frac{d}{dt}\right|_{t=0}\gamma(t),\nabla_tv(0)\right),$$ where the covariant derivative is taken along the path. However, I am not really sure how to start to prove this statement. Does anyone have any ideas or hints?