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.)