Abstract. Let L ⊣ R: X → Y be an adjunction with R monadic and L comonadic. Denote the induced monad on Y by M and the induced comonad on X by C. We characterize those C such that M satisfies the Explicit Basis property. We also discuss some new examples and results motivated by this characterization. 1. The Explicit Basis and Redundant Coassociativity properties In May 2010 Lawvere conjectured that the unit law implies the associative law for comonads arising from EB monads as defined in [14]. The present paper grew out of the intention to understand that conjecture. Let C = (C, ε, δ) be a comonad on a category X. 1.1. Definition. A pre-coalgebra is a pair (X, s) where s: X → CX is a map in X such that the diagram below X s �� CX commutes. id (Of course, pre-coalgebras are just ‘coalgebras for the co-pointed endofunctor (C, ε)’; but we will need to consider both coalgebras and pre-coalgebras for the comonad C and, for this, it is more efficient to have a different name.) Now fix an adjunction L ⊣ R: X → Y with unit η: Id → RL and counit ε: LR → Id. Let C = LR: X → X and denote the induced comonad on X by C = (C, ε, δ). Every pre-coalgebra (X, s) induces a coreflexive pair RX ηR