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I was actually not sure about asking this question since I think I know what the answer is, but here it goes:

Let $M$ and $N$ be two smooth manifolds and $\mathbf{X}$ a vector field defined on $M$. As a function, $\mathbf{X}$ is defined on $M$ and at $p \in M$ it can take values on $T_p M$; in fancier terms, it is a cross section of the tangent bundle of $M$. Now let $\phi: M \to N$ be a smooth function (not necessarily a diffeomorphism) and $\phi_*$ its differential. What kind of object is $\phi_* \mathbf{X}$?

Here are my thoughts: it's not really a vector field on $M$ or on $N$; rather, it's defined on $M$, but at $p \in M$ it takes values in $T_{\phi(p)} N$. Therefore, it's a cross section of a vector bundle with base space $M$ and tangent spaces on $N$ as fibres. At each point $p$ of $M$ we would place the tangent space $T_{\phi(p)}N$. Is this correct? I have a feeling my definition of the vector bundle is not very rigorous; how can it be made more precise? Is there a neater way to define this vector bundle?

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This is usually called (a bit tautologically in this special case) "a vector field along $\phi$", and it is a section in the pullback bundle $\phi^* TN$. See this wikipedia article for a more comprehensive discussion

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