If $\mathbf{C}$ is a category, then the *Yoneda functor* which sends $a$ to $Hom_\mathbf{C}(-,a)$ is a fully faithful embedding of categories
$$ \mathbf{C}\rightarrow \mathbf{Func}(\mathbf{C}^{op},\mathbf{Set})$$

Given any subcategory $\mathbf{B}\subseteq \mathbf{C}$, there is a similar functor $$ \mathbf{C}\rightarrow \mathbf{Func}(\mathbf{B}^{op},\mathbf{Set})$$ which restricts $Hom_\mathbf{C}(-,a)$ to arguments in $\mathbf{B}$.

This functor need not be an embedding in general, but there are many examples where it is, and where its an interesting statement that it is. Examples:

- The category of finite sets, inside the category of sets; or more generally, any subcategory of set containing a non-empty set will work.
- The category of sets and maps which factor through a finite set, inside the category of sets (so $\mathbf{B}$ need not be full).
- The category of affine schemes (ie, $\mathbf{Comm}^{op}$), inside the category of schemes.
- The category of open subsets of $\mathbb{R}^n$ and smooth maps between them, inside the category of smooth $n$-dimensional manifolds.
- The category of abelian groups, inside the category of groups.

My question is, is there a **name** or a nice **characterization** of this subcategories? I'm writing up some notes for a class this semester, and I want to make a remark to this effect. In the absence of a good name, I was going to call them `Yoneda subcategories'.