So I've read (for instance in the introduction to R.S de Jong's [thesis](https://www.math.leidenuniv.nl/~jongrsde/publications/thesis.pdf) ) that the naive adaptation of the proof of the Mordell conjecture over function fields fails, even using Arakelov intersection theory. Most notably we lack a "good" canonical class inequality, for instance Bost, Mestre and Moret-Bailly showed in [this paper](http://www.numdam.org/item/AST_1990__183__69_0.pdf) that the analogue of Bogomolov-Miyao is false.

I was wondering if someone could explain the "proof" of Mordell which would rely on this inequality? I might well be explained in the Bost,Mestre and Moret-Bailly paper, but my french is not really up to the task....