Matthew almost wrote:

> So we just need the category to have arbitrary meets if we want to define \\(\mathcal{Y}^\mathcal{X}\\). For big-boy enriched categories, using monoidal categories rather than monoidal preorders for \\(\mathcal{V}\\), then I suspect if they are *complete* there's an analogue to the construction above.

That sounds right. Say I want a \\(\mathcal{V}\\)-enriched category \\(\mathcal{Y}^\mathcal{X}\\). The objects should be \\(\mathcal{V}\\)-enriched functors and there's no problem defining the set of those if \\(\mathcal{X}\\) and \\(\mathcal{Y}\\) are small (i.e. have a set of objects, rather than a proper class). But then given two \\(\mathcal{V}\\)-enriched functors \\(F, G : \mathcal{X} \to \mathcal{Y}\\) we need to define the \\(\mathcal{V}\\)-object of all \\(\mathcal{V}\\)-natural transformations from \\(F\\) to \\(G\\). That's where the fun starts.

Starting naively, note that a \\(\mathcal{V}\\)-natural transformation from \\(F\\) to \\(G\\) assigns to each object \\(x \in \mathrm{Ob}(\mathcal{X})\\) a morphism from \\(F(x)\\) to \\(G(x)\\), in a such a way that a bunch of naturality squares commute. These naturality square equations can be imposed by taking a limit in \\(\mathcal{V}\\) - since you can impose a bunch of equations using a bunch of equalizers, and 'a bunch of equalizers' is an example of a limit.

But less naively, we don't want to talk about 'a \\(\mathcal{V}\\)-natural transformation from \\(F\\) to \\(G\\)' because we're not shooting for a _set_ of these: we're trying to get a \\(\mathcal{V}\\)-object of these. To build this \\(\mathcal{V}\\)-object we start by taking the product over all \\(x \in \mathrm{Ob}(\mathcal{X})\\) of \\(\mathcal{Y}(F(x),G(x))\\). Then we impose the naturality square equations by taking a suitable limit. That picks out a subobject of

\[ \prod_{x \in \mathrm{Ob}(\mathcal{X})} \mathcal{Y}(F(x),G(x)) \]

and this will be our object

\\[ \mathcal{Y}^\mathcal{X} (F,G) \\]

namely, the object of all \\(\mathcal{V}\\)-natural transformations from \\(F\\) to \\(G\\).

We still need to define composition and such to make \\(\mathcal{Y}^\mathcal{X} \\) into an enriched category. But that should be 'routine'. The key realization is that we need \\(\mathcal{V}\\) to have limits to get \\(\mathcal{Y}^\mathcal{X} \\) up and running, together with whatever assumptions we needed to define \\(\mathcal{V}\\)-categories in the first place. So, assuming \\(\mathcal{V}\\) to be symmetric monoidal and complete sounds good.

(Monoidal may be enough, 'symmetric' may not be necessary here, but in the classic introductions like [Kelly's book](http://www.tac.mta.ca/tac/reprints/articles/10/tr10.pdf) I don't think they try to [remove this particular leg from the centipede](https://en.wikipedia.org/wiki/Centipede_mathematics). Hmm, now that I think about it, we need \\(\mathcal{V}\\) to be symmetric or at least braided to make the _product_ of \\(\mathcal{V}\\)-categories into a \\(\mathcal{V}\\)-category. Since we are now trying to define the exponential \\(\mathcal{Y}^\mathcal{X} \\), which is adjoint to the product, I wouldn't be surprised if \\(\mathcal{V}\\) need to be braided here too! Doing the 'routine' work I mentioned may require moving some objects of \\(\mathcal{V}\\) past each other.)