Representable natural transformations

Stream: maths

[Trebor Huang]

A representable natural transformation $\pi : E\to B$ between two presheaves is such that for every $a : y(X) \to B$, the pullback is again representable.

I'm using this in my project, and I think mathlib doesn't have this yet. What's the right generality to set this up? A brief scan of literature suggests the slightly more general definition: replace the Yoneda embedding with an arbitrary full subcategory $\mathcal C \hookrightarrow \mathcal D$, and call a morphism in $\mathcal D$ to be relatively representable w.r.t. $\mathcal C$.