I have a doubt in this lemma from the book Characteristic classes by Milnor and Stasheff.
Note that $C^{\infty}(\tau^{*}_\mathbb{C} \otimes \zeta) \cong C^{\infty}(\tau^{*}_\mathbb{C}) \otimes_{C^{\infty}(M,\mathbb{C})} C^{\infty}(\zeta)$ and $C^{\infty}(\Lambda^2 \tau^{*}_\mathbb{C} \otimes \zeta) \cong C^{\infty}(\Lambda^2 \tau^{*}_\mathbb{C}) \otimes_{C^{\infty}(M,\mathbb{C})} C^{\infty}(\zeta)$.
If I have defined my map $\widehat{\nabla}$ on a pure tensor $\theta \otimes s$, then (by the above isomorphism) I know the value of $\widehat{\nabla}$ on any other element of $C^{\infty}(\tau^{*}_\mathbb{C} \otimes \zeta)$.
Hence the authors have given the complete description of the map $\widehat{\nabla}$. So what does the uniqueness of $\widehat{\nabla}$ mean in this lemma? And what is the proof of the lemma is trying to do?