Given a manifold $M$.
Denote a chart by $\kappa$.
Introduce the directional derivative: $$\partial:\mathbb{R}^n_a\to T_a\mathbb{R}^n:v\mapsto\partial_v\rvert_a$$
That is an isomorphism with inverse being: $$\varepsilon:T_a\mathbb{R}^n\to\mathbb{R}^n_a:\delta_a\mapsto\oplus_{k=1}^n\delta_a(\pi_k)$$
Is there a short and elegant way to express these maps?
I'm thinking of something like e.g. differentials: $$(D\kappa)_p:T_pU\to T_\hat{p}\hat{U}$$ $$(D\iota)_p:T_pA\to T_pM$$ $$(D\pi_M)_{(p,q)}:T_{(p,q)}(M\times N)\to T_pM$$
As there was nothing better coming up let me close this tread by noting...
The first map is nothing but the directional derivative.
The second map is at most an evaluation of projections.