Finite Limits and Anti-unification in Substitution Categories

It is well-known that coequalisers and pushouts of substitutions correspond to solutions of unification problems, and therefore do not always exist. But how about equalisers and pullbacks? If the literature contains the answers, they are well-hidden.We provide explicit details and proofs for these constructions in categories with substitutions as morphisms, and in particular work out the details of categorial products for which the universal arrow construction turns out to correspond exactly to anti-unification.