Let $\mathcal{K}$ be a $\lambda$-accessible category and $\text{Pres}_{\lambda}(\mathcal{K})$ be the small category of its $\lambda$-presentables.
- It is well known that $\mathcal{K}$ is the $\lambda$-directed cocompletion of $\text{Pres}_{\lambda}(\mathcal{K})$.
Now, call
- $\hat{\mathcal{K}}$ the free directed cocompletion of $\mathcal{K}$.
- $\widehat{\text{Pres}_{\lambda}(\mathcal{K})}$ the free directed cocompletion of $\text{Pres}_{\lambda}(\mathcal{K})$.
Q:
$\widehat{\text{Pres}_{\lambda}(\mathcal{K})} \cong \hat{\mathcal{K}} \ \ ? $
Does the inclusion $\mathcal{K} \hookrightarrow \hat{\mathcal{K}}$ preserve directed colimits?
I expect the answer to the previous question to be $\textit{no}$. Is there a category $\mathcal{D}$ which is a directed cocompletion of $\mathcal{K}$ and such that the inclusion preserves directed colimits?