TITLE: Freely adding finite limits preserves some colimits?
QUESTION [6 upvotes]: Let $\mathcal{K}$ be a category and $\mathcal{K}_{\text{fin}}$ its free completion with finite limits.


Does the embedding $\mathcal{K} \hookrightarrow \mathcal{K}_{\text{fin}}$ preserve some colimits? 


I am especially interested in directed colimits.

REPLY [8 votes]: Yes, all the colimits that exist in ${\cal K}$. Indeed, ${\cal K}_{{\rm fin}}$ can be identified with the smallest full subcategory of ${\rm Fun}({\cal K},{\rm Set})^{{\rm op}}$ which contains the representable functors ${\rm Hom}(x,-)$ and is closed under finite limits, and the composition ${\cal K} \to {\cal K}_{{\rm fin}} \hookrightarrow {\rm Fun}({\cal K},{\rm Set})^{{\rm op}}$ is the opposite of the Yoneda embedding ${\cal K}^{{\rm op}} \to {\rm Fun}({\cal K},{\rm Set})$. The result you need now follows from the fact that the Yoneda embedding preserves all limits which exist in ${\cal K}^{{\rm op}}$.