Idempotent completion of cubes in posets

This note concerns the category $\mathrm{\square }$ of cartesian cubes with connections, equivalently the full subcategory of posets on objects $[1{]}^n$ with $n\ge 0$. We show that the idempotent completion of $\mathrm{\square }$ consists of finite complete posets. It follows that cubical sets, ie presheaves over $\mathrm{\square }$, are equivalent to presheaves over finite complete posets. This yields an alternative exposition of a result by Kapulkin and Voevodsky that simplicial sets form a subtopos of cubical sets.