This question is a follow-up of Extending functors defined on dense subcategories.

Let $\mathcal{K}$ be a locally presentable category. An object $X$ of $\mathcal{K}$ is called a

Yoneda objectif 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 $\text{Set}^\mathcal{A}$ ?