@Anindya,

My bad! I started to use the notation \$$\Rightarrow\$$ Hinze uses for functors in his paper [Kan Extensions for Program Optimisation](https://www.cs.ox.ac.uk/ralf.hinze/Kan.pdf). I'll leave the typo there and just stop using that notation.

Regarding the enriched Yoneda lemma, [Hinich (2016)](http://www.tac.mta.ca/tac/volumes/31/29/31-29.pdf) proves the lemma for \$$\mathcal{M}\$$-enriched categories where \$$\mathcal{M}\$$ is a monoidal category with colimits. I am not sure if this is stronger than the theorem your friend was alluding to, or there are just multiple conditions for enriched categories that entail the Yoneda lemma.