In Lectures on Integrability of Lie Brackets, while defining the concept of a non-base preserving Lie algebroid, Crainic and Fernandes claim the following (beginning of page 27):
Given vector bundles $A_1 \to M_1, A_2 \to M_2$, let there be a vector bundle morphism $F : A_1 \to A_2$ over a map $f : M_1 \to M_2$. Also, take any section $\alpha \in \Gamma(A_1)$ and consider the section $F( \alpha)$ of the pullback bundle $f^* A_2 \to M_1$. Then there are finitely many functions $c_i \in C^\infty(M_1)$ and sections $\alpha_i \in \Gamma(A_2)$ so that $$ \sum_i c_i \cdot (\alpha_i \circ f) = F(\alpha).$$
After some investigation, I found that Higgins and Mackenzie in Algebraic Constructions in the Category of Lie Algebroids elaborate on this statement (at the beginning of Section 1) by claiming that $\Gamma(f^*A_2) \cong C^\infty(A_1) \otimes_{C^\infty(A_2)} \Gamma(A_2)$ and citing yet another book for this statement, which I do not have access to. Since Crainic and Fernandes claimed the statement without much elaboration, my questions is: Is there a simple perspective to look at this problem which I am not seeing, or an easy proof? Perhaps as a clever consequence of Serre-Swan or something?
Thought it might be worth asking :) Thank you!