I am struggling to understand the proof of lemma 6 in Serre's GAGA.
why does coherence of $\mathscr{F}$ imply Hom($\mathscr{F}^{h}$,$\mathscr{G}^{h}$)= Hom($\mathscr{F}_x$$\otimes$$\mathscr{H}_x$,$\mathscr{G}_x$$\otimes$$\mathscr{ H}_x$) where $\mathscr{F}^h$ is defined as definition 2 in paper.
and why is $i_x $: Hom($\mathscr{F}_x$,$\mathscr{G}_x$)$\otimes$$\mathscr{H}_x$$\rightarrow$Hom($\mathscr{F}_x$$\otimes$$\mathscr{H}_x$,$\mathscr{G}_x$$\otimes$$\mathscr{ H}_x$) bijective provided that ($\mathcal{O}_x$, $\mathcal{H}_x$) is a flat couple (meaning $\mathcal{H}_x$/$\mathcal{O}_x$ is $\mathcal{O}_x$ flat).
you can find GAGA here: https://math.berkeley.edu/~reb/courses/256A/
any hints to just some of the above questions would be greatly appreciated.