My book is An Introduction to Manifolds by Loring W. Tu.
Does Problem 18.8 below follow from an earlier exercise?
Problem 18.8 (Pullback by a surjective submersion)
Problem 10.5a (Injectivity of the dual map)
My thought is that just as we have a dual for k-covectors:
So do we have a dual for k-forms, although that might be what this exercise is trying to prove.
One reason I think Problem 10.5a doesn't apply is that that $\pi$ is a submersion is used to prove $\pi^{*}$ is injective (see below), while Problem 10.5a does not seem to say anything explicitly about submersions. Either that or the proof below is just a proof that is alternative, direct and doesn't rely on category theory.
Problem 18.8 can be proven in the following way (where I hope to not miss anything important hiding under the surface; let me know if you find issues in the proof!).
You should first note that the pullback map $\pi^*$ is the dual map to the map $\pi$. Now, a general fact from linear algebra is that for any linear map $f$, $$f\text{ injective } \Longleftrightarrow f^*\text{ surjective,}\\ f\text{ surjective } \Longleftrightarrow f^*\text{ injective.} $$ So in your case, if $\pi$ is surjective, $\pi^*$ is an injective homomorphism of vector spaces and you are left showing that it respects the algebra structure on $\Omega^*(M)$, i.e. that everything fits well with the differential. I think that you will need the property of being a submersion in that. Try to proceed with what I wrote, if you can't, I will work out a solution and put it here.