Let $E \to N$ be a smooth vector bundle of rank $r$, let $f: M \to N$ be smooth, and let $0 \le k \le r$. There is a canonical vector bundle isomorphism $f^*(E^*) \cong (f^*E)^*$. There is also a canonical vector bundle isomorphism $f^*(\Lambda^k E) \cong \Lambda^k(f^*E)$. The dual isomorphism sends $(x,\lambda)$ to the functional on $(f^*E)_x$ defined by $(x,e) \mapsto \lambda(e)$.