Let $\mathcal{K}$ be a $\lambda$-accessible category and $\hat{\mathcal{K}}$ its free completion under connected limits. 


* Is $\hat{\mathcal{K}}$ still accessible?
* $\mathcal{K}$ can be identified with a subcategory of $\text{Set}^{\text{Pres}\_{\lambda}(\mathcal{K}) ^{\text{op}}}$, is it true that $\hat{\mathcal{K}}$ is just the closure under connected limits of this subcategory?


It looks like the answer should be *no* to both questions, especially the second one. On the other hand, I am interested in some counterexamples. Still I have not properly understood the phenomenon of (co)completion. 



Freely adding connected limits preserves accessibility?
An easy way to see the failure of (2) is that if $\mathcal{K}$ already has connected limits, then its restricted Yoneda embedding into $\mathrm{Set}^{\mathrm{Pres}\_\lambda(\mathcal{K})^{\mathrm{op}}}$ preserves them, so that its closure under such limits therein would be just itself. But the *free* completion under connected limits has to add *new* limits that do not coincide with the ones that already existed.


