This is the main question:

if $p:A \rightarrow B$ is a smooth vector bundle homomoprhism over base space $M$, then $pX$ is a smooth section of $B$, where $X \in \Gamma(A)$ is a smooth section of $A$.

I think the claim is simply true in local coordinates, being composition of smooth maps.

If the above claim is not true. Here is more context. I am reading page 10, Definition 1.5 of the notes which defines a Lie algebroid.

let $M$ be a smooth manifold, and let $p:A \rightarrow TM$ be a bundle homomorphism. Then we require it to satisfy, $$ [pX,pY]_{TM}= p[X,Y]_A$$ $X,Y \in \Gamma(A)$, the space of smooth sections. The Lie brackets are with respect to the spaces $\Gamma(TM)$ and $\Gamma(A)$.

My question is, how are the terms well defined - is it guaranteed that $pX$ is still a smooth vector field over $M$?


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