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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.