- The property of a category studied in the paper can be seen as a natural weakening of the property that all regular epimorphisms are normal epimorphisms. Pointed varieties having this latter property were called
*varieties with ideals*by K. Fichtner, who obtained their syntactical characterization in 1968. Pointed regular categories having this property have been called*normal categories*in my paper in 2010. This term was independently suggested by my father, G. Janelidze. In universal algebra, there is a non-pointed version of this notion for which the term “0-regular variety” is used. - I showed in my PhD thesis that all product projections $A\times B\rightarrow B$ are normal if and only if the product projections $A\times A\rightarrow A$ are normal.
- It is established in the paper that all pointed subtractive varieties and all Jónsson-Tarski varieties have normal projections. This means that pointed categories of essentially all structures of interest to algebra have normal projections. A counterexample if given by the category of pointed sets.
- In my second talk at the Australian Category Seminar in 2003, where I showed that unital categories have normal projections. G. M. Kelly remarked during the talk that what now are called weakly unital categories also have normal projections. The notion of a weakly unital category is a N. Martins-Ferreira’s style of generalization of the notion of a unital category, and it was first studied in the PhD thesis of J. R. A. Gray.