# Pushforward of smooth section is smooth?

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 $$M$$?