= Solution
At $p\in\mathcal Q$, the <differential of a smooth map> defines the <pushforward of a vector field> by
$$
\boxed{(\psi_*X)_p=d\psi_p(X_p),\qquad ((\psi_*X)_p f)=X_p(f\circ\psi),\quad f\in C^\infty(N).}
$$
This is a <tangent vector> at $\psi(p)$, and varies smoothly as a section of the <pullback tangent bundle>. In <local coordinates> its components are $X^i(p)\partial_i\psi^a(p)$.
For a general <smooth map between manifolds>, this is a <vector field along a map>, rather than an intrinsic vector field on all of $N$. A vector field $Y$ on the image can be defined only if these vectors agree whenever two points have the same image, and the resulting field must also be smooth; a <projectable vector field> must meet that requirement. For a <diffeomorphism>, there is no ambiguity and $Y_q=d\psi_{\psi^{-1}(q)}X_{\psi^{-1}(q)}$. For example, $\psi(x)=x^2$ on $\mathbb R$ and $X=\partial_x$ give $2x\partial_y$, with opposite nonzero values at the two preimages of $y>0$. Thus the source's notation must be read with this qualification.
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