Let $\mathcal{K}$ be a category and $\mathcal{K}\_{\text{fin}}$ its free completion with finite limits.



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> * Does the embedding $\mathcal{K} \hookrightarrow \mathcal{K}\_{\text{fin}}$ preserve some colimits?
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I am especially interested in directed colimits.



Freely adding finite limits preserves some colimits?
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 ${\cal K}^{{\rm op}}$. 


