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This is from Lee's Introduction To Smooth Manifolds Second Edition, page 76. Should $\Phi: T_pM \rightarrow \mathcal{D}_pM$, not $\Phi: \mathcal{D}_pM \rightarrow T_pM$? Just looking at how $\Phi$ is defined, it looks like it is producing a derivation on a germ.

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The notation can be confusing, but actually, I do not think this is a typo. $v$ is derivation on a germ ($v \in \mathcal{D}_pM$) and $\Phi$ transforms $v$ into a derivation of a smooth function ($\Phi v \in T_pM$) as follows: The value of the smooth function derivation $\Phi v$ at a smooth function $f$ is defined to be equal to the value of the germ derivation $v$ on the germ $[f]_p$.

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    $\begingroup$ That's exactly right. $\endgroup$ – Jack Lee Oct 15 '18 at 19:19

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