Distributive semilattices as retracts of ultraboolean ones; functorial inverses without adjunction

A left angle bracketlogical or,0right-pointing angle bracket-semilattice is ultraboolean, if it is a directed union of finite Boolean left angle bracketlogical or,0right-pointing angle bracket-semilattices. We prove that every distributive left angle bracketlogical or,0right-pointing angle bracket-semilattice is a retract of some ultraboolean left angle bracketlogical or,0right-pointing angle bracket-semilattice. This is established by proving that every finite distributive left angle bracketlogical or,0right-pointing angle bracket-semilattice is a retract of some finite Boolean left angle bracketlogical or,0right-pointing angle bracket-semilattice, and this in a functorial way. This result is, in turn, obtained as a particular case of a category-theoretical result that gives sufficient conditions, for a functor Π, to admit a right inverse. The particular functor Π used for the abovementioned result about ultraboolean semilattices has neither a right nor a left adjoint.