  Motivated by the categorical-algebraic analysis of split epimorphisms of
monoids, we study the concept of a special object induced by the intrinsic
Schreier split epimorphisms in the context of a regular unital category with
binary coproducts, comonadic covers and a natural imaginary splitting in the
sense of our article [Intrinsic Schreier split extensions, Appl. Categ.
Structures 28 (2020), 517--538]. In this context, each object comes naturally
equipped with an imaginary magma structure. We analyse the intrinsic Schreier
split epimorphisms in this setting, showing that their properties improve when
the imaginary magma structures happen to be associative. We compare the
intrinsic Schreier special objects with the protomodular objects, and
characterise them in terms of the imaginary magma structure. We furthermore
relate them to the Engel property in the case of groups and Lie algebras.
