Matthew and Christopher wrote:

> If we have a pair \$$f \dashv g: \mathcal{X} \to \mathcal{Y}\$$ of adjoint short maps then can we construct an equivalence between \$$\mathcal{X}\$$ and \$$\mathcal{Y}\$$?

That's a nice question! But the first thing to note is that we haven't officially discussed adjoint functors between enriched categories! That's just due to a shortage of time.

So, the first thing to do is figure out the definition of adjoint \$$\mathcal{V}\$$-functors between \$$\mathcal{V}\$$-categories when \$$\mathcal{V}\$$ is a symmetric monoidal preorder. Or look it up, but figuring it out is much better.
We know what \$$\mathcal{V}\$$-functors are - I explained it in this lecture. So we just need to make up a concept of adjoint \$$\mathcal{V}\$$-functors that generalizes the concept of adjoint monotone maps between preorders, which is the special case \$$\mathcal{V} = \mathbf{Bool}\$$.

Once we see the definition, so we know what the question actually _is_, I bet we can answer the question without a huge amount of work.

So:

**Definition.** Given \$$\mathcal{V}\$$-categories \$$\mathcal{X}\$$ and \$$\mathcal{Y}\$$, we say a \$$\mathcal{V}\$$-functor \$$F : \mathcal{X} \to \mathcal{Y}\$$ is the **left adjoint** of a \$$\mathcal{V}\$$-functor \$$G : \mathcal{Y} \to \mathcal{X}\$$ if....