[My Question] Why "$F$ is flat is needed in this proof?
[My attempt]
Let $f : \mathcal{a} \otimes F \rightarrow \mathcal{a}F $ be defined by $f(\sum_i x_i \otimes b_i) = \sum_ix_i b_i$ . (By universal property, this map exists)
Also let $g : \mathcal{a}F \rightarrow \mathcal{a} \otimes F $ be defined by $g(\sum_ix_i b_i) = \sum_i x_i \otimes b_i$ .
Then I think $f \circ g = i_d$ and $ g \circ f = i_d$ ,so they are isomorphic.
Where is the problem in this proof?
Thanks.