Matthew wrote approximately:

> Regarding the enriched Yoneda lemma, [Hinich (2016)](http://www.tac.mta.ca/tac/volumes/31/29/31-29.pdf) proves the lemma for \\(\mathcal{V}\\)-enriched categories where \\(\mathcal{V}\\) 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.

I'm pretty sure with different assumptions on your 'base enrichment' (the thing I'm calling \\(\mathcal{V}\\)), you get different ways of stating the Yoneda idea, and different versions of the Yoneda lemma and Yoneda embedding.

It's surprising that Hinich's version doesn't assume \\(\mathcal{V}\\) is symmetric or closed, but his paper only dates back to 2016, and he makes a big deal about not needing these assumptions, so clearly it counts as a bit of a novelty to figure out how to even state the Yoneda idea in this generality. (He does however say it was already done by Garner and Shulman in another 2016 paper.)

For example, the Yoneda embedding is usually a functor from your \\(\mathcal{V}\\)-category \\(\mathcal{C}\\) into the 'presheaf category' \\(\mathcal{V}^{\mathcal{C}^{\text{op}}}\\), but we've seen \\(\mathcal{C}^{\text{op}}\\) isn't a \\(\mathcal{V}\\)-category unless \\(\mathcal{V}\\) is symmetric (or for us, commutative). I see that Hinich gets around this using the fact that \\(\mathcal{C}^{\text{op}}\\) is a \\(\mathcal{V}^{\text{op}}\\)-category!

However, it's important to know that all this is [centipede mathematics](https://en.wikipedia.org/wiki/Centipede_mathematics). Usually people are happiest doing enriched category theory when \\(\mathcal{V}\\) is closed symmetric monoidal with colimits. Then you can do stuff without worrying too much... and most of the really important examples fit in this framework.