Rigidity of unary algebras and its application to the $${\mathcal {HS} = \mathcal {SH}}$$ problem.

H. P. Gumm and T. Schröder stated a conjecture that the preservation of preimages by a functor T for which | T1| = 1 is equivalent to the satisfaction of the class equality $${{\mathcal {HS}}({\sf K}) = {\mathcal {SH}}({\sf K})}$$ for any class K of T-coalgebras. Although T. Brengos and V. Trnková gave a positive answer to this problem for a wide class of Set-endofunctors, they were unable to find the full solution. Using a construction of a rigid unary algebra we prove $${{\mathcal {HS}} \neq {\mathcal {SH}}}$$ for a class of Set-endofunctors not preserving non-empty preimages; these functors have not been considered previously. [ABSTRACT FROM AUTHOR]