Let $\mathrm{Ind}(\mathcal{C})$ be the Ind-completion. We can define it in two different (but similar) ways: as filtered colimits of representable presheaves and as the category of diagrams over filtered categories: https://ncatlab.org/nlab/show/ind-object It is easy to show for the first definition, using the Yoneda lemma and the fact that colimits of presheaves are computed objectwise, that every object of $\mathcal{C}$ under Yoneda embedding is compact in $\mathrm{Ind}(\mathcal{C})$.
However I can't see why it is true for the second definition as well. Here object $X$ of $\mathcal{C}$ in $\mathrm{Ind}(\mathcal{C})$ is the diagram that looks like $$X: {1}\to \mathcal{C}, X(*)=X$$ And morphisms are defined as $$\mathrm{Ind}(\mathcal{C})(F, G)=\lim_i \mathrm{colim}_j \mathrm{Hom} _{\mathcal C} (F(i), G(j)) $$
For compactness of $X$ we need to show that $$\mathrm{colim}_j \mathrm{Hom} _{\mathcal C} (X, Y_k(j)) \cong\lim_i \mathrm{colim}_j \mathrm{Hom} _{\mathcal C} (X(i), Y_k(j))\cong\mathrm{Ind}(\mathcal{C})(X,\mathrm{colim}_k Y_k)\cong\mathrm{colim}_k \mathrm{Ind}(\mathcal{C})(X, Y_k)\cong\mathrm{colim}_k\lim_i \mathrm{colim}_j \mathrm{Hom} _{\mathcal C} (X(i), Y_k(j))\cong \mathrm{colim}_k\mathrm{colim}_j \mathrm{Hom} _{\mathcal C} (X, Y_k(j))$$ Here $i\in Ob(1)=\{ * \}$ and $\mathrm{colim}_k$ is filtered. I suppose the proof should be quite easy, but I can't see it at all.