I have posted this problem recently and I think that as it was too long, not many people took interest in it. So I am reposting it also I am giving you an earlier link :I had a problem in manifold which states:
Show that $A^1:Man^{op} \to Mod$ is a functor, where $M$ is a manifold, $A^1(M)$ is the $1$-form on $M$. and for $\phi:M \to N$, $A^1(\phi)=\phi^*$. This means to show that
a) $A^1(M)$ is a module.(What is the ring?)
b) $\phi^*$ preserves module operations(does the ring change?)(I feel yes! but not getting why!!)
c)$(1_M)^*=1_{A^1(M)}$
In part c) I feel that I have to show $w \circ (\Bbb 1_M)_*=(\Bbb 1_{A^1(M)}w$ as $w \circ (\Bbb 1_M)_*=(\Bbb 1_M)^*w$ but how!! Please help me in a), b) and c). Especially I feel that the ring in a) should be $\Bbb R$