  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 $X\times Y$ 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.
