This paper presents a systematic study of coproducts. This is carried outprincipally, but not exclusively, for finitely generated quasivarieties A thatadmit a (term) reduct in the variety D of bounded distributive lattices. Inthis setting we present necessary and sufficient conditions on A for theforgetful functor U from A to D to preserve coproducts. We also investigate thepossible behaviours of U as regards coproducts in A under weaker assumptions.Depending on the properties exhibited by the functor, different procedures arethen available for describing these coproducts. We classify a selection ofwell-known varieties within our scheme, thereby unifying earlier results andobtaining some new ones. The paper's methodology draws heavily on dualitytheory. We use Priestley duality as a tool and our descriptions of coproductsare given in terms of this duality. We also exploit natural duality theory,specifically multisorted piggyback dualities, in our analysis of the behaviourof the forgetful functor into D. In the opposite direction, we reveal that thetype of natural duality that the class A can possess is governed by propertiesof coproducts in A and the way in which the classes A and U(A) nteract.
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