Profinite completions of products

A source of difficulty in profinite homotopy theory is that the profinite completion functor does not preserve finite products. In this note, we provide a new, checkable criterion on prospaces X and Y that guarantees that the profinite completion of X× Y agrees with the product of the profinite completions of X and Y. Using this criterion, we show that profinite completion preserves products of \'{e}tale homotopy types of qcqs schemes. This fills a gap in Chough's proof of the K\"{u}nneth formula for the \'{e}tale homotopy type of a product of proper schemes over a separably closed field.