Canonical transformation $T_pV \cong V$?

Question

Apparently there is a natural isomorphism between a vector space and the tangent space of this vector space at one point. I.e., $\forall p \in V$ we have $T_pV \cong V$. I know that the isomorphism is $v \mapsto D_{v\vert a}$. But what is the isomorphism explicitely in the other direction? If I take a derivation $X\in T_p V$ to what vector $v \in V$ is it sent?

Answer 1

The inverse isomorphism is given by $$X \mapsto \exp_p(X) - p$$