Strengthening the hypotheses on $P$ and $Q$ from a fixed point *property* (every endofunction has *some* fixed point) to the existence of a fixed point *operator* $\mathsf{fix}$) with
$$ \text{for any } f:P\to P, \qquad f({\mathsf{fix}}(f)) = {\mathsf{fix}}(f), $$
there is a standard theorem in **domain theory** or **lambda calculus** (theoretical computer science) due to **Hans Bekić**.