Jim: I should know what you're talking about, that but I don't. In fact I don't believe that functor has an adjoint unless \\(\mathcal{C}\\) is "sufficiently nice", e.g. a topos or something.

You didn't say what functor

\[ \mathcal{C} \to \mathbf{Set}^{\mathcal{C}^{\text{op}}} \]

he's talking about; you can imagine various ones, but the best one is the Yoneda embedding. Everyone who wants to understand that should read this:

* Tae Danae-Bradley, [The Yoneda embedding](http://www.math3ma.com/mathema/2017/9/6/the-yoneda-embedding).

You didn't say what functor

\[ \mathcal{C} \to \mathbf{Set}^{\mathcal{C}^{\text{op}}} \]

he's talking about; you can imagine various ones, but the best one is the Yoneda embedding. Everyone who wants to understand that should read this:

* Tae Danae-Bradley, [The Yoneda embedding](http://www.math3ma.com/mathema/2017/9/6/the-yoneda-embedding).