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.