For a Riemannian manifold $M$, and each $x\in M$, we can define the exponential map
$$\exp_x:T_xM\to M.$$
Then for each vector $v\in T_x M$, we have the differential
$$(d\exp_x)_v:T_vT_xM \to T_{\exp_x(v)}M.$$
Why do people say $(d\exp_x)_v$ is the identity map? I understand that we can identify $T_vT_xM$ with $T_xM$ by a translation $u\mapsto u-v$. But $T_xM$ and $T_{\exp_x(v)}M$ are completely different vector spaces!