Freely adding finite limits preserves some colimits?

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.

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 ${\cal K}^{{\rm op}}$.
• @IvanDiLiberti, for a category ${\cal C}$, the Yoneda embedding ${\cal C} \to {\rm Fun}({\cal C}^{{\rm op}},{\rm Set})$ sends $x \in {\cal C}$ to ${\rm Hom}(-,x): {\cal C}^{{\rm op}} \to {\rm Set}$. Now if $x$ is the limit of a diagram $\{x_i\}_{i \in {\cal I}}$ then clearly ${\rm Hom}(-,x) = {\rm lim}_{i \in {\cal I}}{\rm Hom}(-,x_i)$, but if $x$ is the colimit of a diagram then in general there is nothing to be said about maps into $x$. That's why the Yoneda embedding preserves limits and not colimits (or why, in your case, the opposite of Yoneda preserves colimits but not limits). – Yonatan Harpaz Nov 2 '17 at 19:37