Relational Sheaves for a Heyting Algebra

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We show that for a Heyting algebra H, a relational-presheaf is an idempotent symmetric order-preserving lax-semifunctor. A relational-presheaf is a relational-sheaf, if it is an idempotent infima-preserving lax semifunctor. The associated relational-sheaf functor factors through the category of sheaves for H. Using this and the appropriate comparison theorems we obtain the main result that the associated categories of relational-presheaves and relational-sheaves are each respectively equivalent to the categories of presheaves and sheaves for H.