I got a bit stuck while studying through Ravi Vakil's FOAG.
The task is to prove that for an affine morphism $\pi: X\to Y$ and two quasicoherent sheaves $F$ on $X$ and $G$ on $Y$ the natural map
$$(\pi_*F)\otimes G\to\pi_*(F\otimes\pi^*G)$$
is an isomorphism. I constructed the above map through abstract nonsense as the composition
$$(\pi_*F)\otimes G\to(\pi_*F)\otimes\pi_*\pi^*G\to\pi_*(F\otimes\pi^*G),$$
where the left map comes from the counit of the adjunction, and the right map comes from the "sheafification" map $(\pi_*L)\otimes (\pi_*S)\to\pi_*(L\otimes S).$
Now for an affine morphism $\pi$, the right map is an isomorphism over any affine open, and since all sheaves are quasicoherent, it's an isomorphism. I am now trying to check that the map $G\to \pi_*\pi^* G$ gives an isomorphism on stalks. However, I don't think there is a nice description of the stalk of pushforward, and the fact that $\pi$ is affine does not seem to simplify matters.
Actually, I am not even sure this statement is true. Consider $X=A^1_k,\ Y=\text{Spec } k,$ $G=\mathcal{O}_Y$. Then $\pi^*(G)=\mathcal{O}_X,$ $\pi_*(\mathcal{O}_X)([0])=\Gamma(X,\mathcal{O}_X)=k[t].$
Is the statement of the problem wrong, or did I make a mistake somewhere?