A remark on fixed points of functors in topological categories (Q1364915)

Let \(F\) be an endofunctor of a concrete category \(\mathcal K\). A cardinal \(\alpha\) is a fixed cardinal of \(F\) if for every \(\mathcal K\)-object \(U\) with an underlying set of cardinality \(\alpha\) the cardinality of the underlying set of \(FU\) is at most \(\alpha\). We say that a \(\mathcal K\)-object \(U\) is a fixed point of \(F\) if there exists an isomorphism \(f:FU@>>> U\). A fixed point \(U\) of \(F\) with an isomorphism \(f:FU@>>>U \) is the least (greatest) fixed point of \(F\) if for every fixed point \(V\) of \(F\) with an isomorphism \(g:FV@>>> V\) there exists exactly one \(\mathcal K\)-morphism \(h:U@>>> V\) (or \(h:V@>>> U\)) with \(g\circ Fh=h\circ f\) (or \(f^{-1}\circ h=Fh\circ g^{-1}\), respectively). If \(\mathcal K\) is a fibre-small monotopological category and if \(F\) preserves bimorphisms then \(F\) has the least fixed point whenever there exists a fixed cardinal of \(F\). If \(\mathcal K\) is a fibre-small topological category and if \(F\) preserves monomorphisms then \(F\) has the greatest fixed point whenever there exists a cardinal \(\alpha\) such that every cardinal \(\beta\) with \(\alpha\leq\beta\leq 2^{\alpha}\) is a fixed cardinal of \(F\).