Anindya wrote:

> I once asked a category theoretician friend of mine what this "special" was exactly, and he replied "symmetric monoidal closed".

That'll get you the weak version of the enriched Yoneda lemma stated here:

* nLab, [Enriched Yoneda Lemma](https://ncatlab.org/nlab/show/enriched+Yoneda+lemma).

They write:

> We discuss here two forms of the Yoneda lemma.

> Let \\(\mathcal{V}\\) be a (locally small) closed symmetric monoidal category, so that \\(\mathcal{V}\\) is enriched in itself via its internal hom.

> ### Weak form

> A _weak form_ of the enriched Yoneda lemma says that given a \\(\mathcal{V}\\)-enriched functor \\(F: \mathcal{C} \to \mathcal{V}\\) and an object \\(c\\) of \\(\mathcal{C} \\), the _set_ of \\(\mathcal{V}\\)-enriched natural transformations \\(\alpha: \mathcal{C} (c, -) \Rightarrow F\\) is in natural bijection with the set of **elements** of (\\(F(c)\\), i.e., the set of morphisms \\(I \to F(c)\\).

The strong form discussed on the nLab assumes also that \\(\mathcal{V}\\) has all (small) colimits. We can use those to define a _\\(\mathcal{V}\\)-object_ of \\(\mathcal{V}\\)-enriched natural transformations \\(\alpha: \mathcal{C} (c, -) \Rightarrow F\\), and prove this is isomorphic to \\(F(c)\\).

Clearly the strong version is nicer in that it moves further away from the world of sets and into the world of \\(\mathcal{V}\\): it gives you the object \\(F(c)\\) instead of merely its set of elements, which in general contains a lot less information.