@DenisNardin I wonder if you remember our discussion on Prop. 4.16 of https://arxiv.org/pdf/1803.10897.pdf .
where I asked whether F preserving finite products was necessary.
Back then i thought we can decompose any henselian unital algebra as its p primary parts I = \oplus I_p. But its not clear why I_p is henseilan. i.e. it doesn't necessarilly satisfy the "solution solving criteria at Definition 3.7. So why are using that F prserves products?
I have a similar problem: for the next llne of the proof. in short exact sequence pI->I->I/pI why is pI also a henselian algebra?