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....

> 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....