AbstractThe varieties of dialgebras (also known as Loday-type algebras) over a given type of algebra have been the subject of multiple recent developments. We provide here a construction of such dialgebra varieties via bimodules over an algebra and a surjective equivariant map. Our construction is equivalent to the KP construction (Kolesnikov–Pozhidaev construction) when departing from the set of linearized identities of the algebra variety. The novel construction simplifies the obtention of the dialgebra equations without forcing a complete linearization of the algebra identities. We illustrate the use of the novel construction providing the dialgebras associated to several varieties of algebras, including those over diverse Lie admissible algebras. We provide some novel explorations on the structure of the dialgebras which are easily articulated through our construction.Keywords: nonassociative rings and algebrasdialgebrasLeibniz algebrasAMS Subject Classifications: 17A0117A3017A32 NotesNo potential conflict of interest was reported by the authors.Additional informationFundingThe first and third authors were supported under Universidad Nacional de Colombia project Álgebras de Leibniz y el problema Coquecigrue, [grant number 19523]; and the second author was supported by Sistema Universitario de Investigaciones (SUI) – Universidad de Antioquia.