This question is a follow-up of [Extending functors defined on dense subcategories](https://mathoverflow.net/q/273164/24563).



> 
> Let $\mathcal{K}$ be a locally presentable category. An object $X$ of
>  $\mathcal{K}$ is called a *Yoneda object* if the functor
>  $\mathcal{K}(X,-):\mathcal{K} \to \text{Set}$ is colimit-preserving.
> 
> 
> 


For example, the representable functors $\mathcal{A}(a,-)$ of $\text{Set}^\mathcal{A}$, where $\mathcal{A}$ is a small category are Yoneda objects since $\text{Set}^\mathcal{A}(\mathcal{A}(a,-),F)=F(a)$. A Yoneda object is finitely presentable.



> 
> Question 1: Are there other Yoneda objects in $\text{Set}^\mathcal{A}$
>  than the representable functors (up to isomorphisms of functors) ?
> 
> 
> 


Every locally presentable category is a reflective subcategory of a category of the form $\text{Set}^\mathcal{A}$.



> 
> Question 2: Let $\mathcal{K}$ be a locally presentable category having
>  a strong generator of Yoneda objects. Is this category equivalent to a
>  category of the form $\text{Set}^\mathcal{A}$ ?
> 
> 
> 



About the Yoneda objects of a locally presentable category
The usual name for "Yoneda objects" is "[tiny](https://ncatlab.org/nlab/show/tiny+object)" or "small-projective".


1. In general the tiny objects in a presheaf category are the *retracts* of representables. In particular, if $A$ is Cauchy-complete, then every tiny object is representable.
2. Yes. In fact, instead of local presentability it is enough to assume cocompleteness; see Theorem 5.26 of [Basic Concepts of Enriched Category Theory](http://www.tac.mta.ca/tac/reprints/articles/10/tr10.pdf).


