Determinisation of relational substitutions in ordered categories with domain

Restating a unification problem as a single relational substitution instead of as multiple functional substitutions (or terms), a solution becomes a “determiniser” arrow and allows formalisation in the context of locally ordered categories with domain. This relies on the determinacy concept of “characterisation by domain” introduced by Desharnais and Möller for Kleene algebras with domain; this is here applied in the weakest possible setting.We show how “most general determinisers” can be seen as generalisation of quotient projections of partial equivalence relations, and show a characterisation that manages to avoid using converse or symmetry by employing restricted residuals instead.