Theorem: Let $\mathcal{C}$ be a category and $\mathrm{Kar}(\mathcal{C})$ be its idempotent completion. Let $I:\mathcal{C}\to \mathrm{Kar}(\mathcal{C})$ be the inclusion functor. Then for any category $\mathcal{D}$ in which idempotents split and any functor $F:\mathcal{C}\to \mathcal{D}$, there is a unique (up to isomorphism) functor $F':\mathrm{Kar}(\mathcal{C})\to \mathcal{D}$ such that $F'\circ I=F$.