I'v asked this question on StackExchange but unfortunately nobody answered. I thought that maybe it would be more apropriate to post it here:
so suppose that we have a morphism $\theta: \Gamma(B,E_1) \to \Gamma(B,E_2)$ where $E_i$ are two vector bundles over $B$ and let $f:A \to B$ be a continuos map. Then we can define a pullback bundles $f^*E_1$ and $f^*E_2$.
Question 1: first of all I would like to know whether one can somehow describe $\Gamma(A,f^*E_1)$ in terms of $\Gamma(B,E_1)$
And then I would also like to know:
Question 2: How a given morphism $\theta$ like above induces morphism $f^*\theta: \Gamma(A,f^*E_1) \to \Gamma(A,f^*E_2)$.
I tried the following argumentation: once we have a morphism $\theta$ like above it produces a morphism of bundles $\varphi:E_1 \to E_2$. This induces a morphism $(x,v) \mapsto (x,\varphi(v))$ between pullback bundles and this gives rise to the morphism on the level of sections. Is there a direct approach, which allows us working only on the level of sections? If so, is it equivalent to described above?