In the notes  [Geometric Wave Equations][1] by Stefan Waldmann at   page 70  they have 


> Having a fixed Lorentz metric $g$ on a spacetime manifold $M$ we can
> now transfer the notions of special relativity, see e.g. 50 , to $(M,
 g)$. In fact, each tangent space $\left(T_{p} M, g_{p}\right)$ is
> isometrically isomorphic to Minkowski spacetime $\left(\mathbb{R}^{n},
\eta\right)$ with $\eta=\operatorname{diag}(+1,-1, \ldots,-1)$, by
> choosing a Lorentz frame: there exist tangent vectors $e_{i} \in T_{p}
 M$ with $i=1, \ldots, n$ such that $$ g_{p}\left(e_{i},
 e_{j}\right)=\eta_{i j}=\pm \delta_{i j} . $$

We say that two manifolds $M$ and $N$ are isometric if we have vectors $v \in T_pM$, $u \in T_{\phi(p)}N$ and a map $\phi:M\rightarrow N$ such that 

$g(v,v)=g'(\phi^*v,\phi^*v)$ where $g$ is a metric in $M$ ,  $g'$ is a metric in $N$ and $\phi^*$ denotes a pushfoward.

Now the definition of isometry  refers  to two manifolds, but in the notes they are claiming an isometry between a manifold and a tangent space. 

How is this isometry constructed?
   
[1]: https://arxiv.org/abs/1208.4706