Classifying categories for partial equational logic

Along the lines of classical categorical type theory for total functions, we establish correspondence results between certain classes of partial equational theories on the one hand and suitable classes of categories having certain finite limits on the other hand. E.g., we show that finitary partial theories with existentially conditioned equations are essentially the same as cartesian categories with distinguished domains, and that partial λ-calculi with internal equality are equivalent to a suitable class of partial cartesian closed categories.