  In proof-theoretic semantics, model-theoretic validity is replaced by
proof-theoretic validity. Validity of formulae is defined inductively from a
base giving the validity of atoms using inductive clauses derived from
proof-theoretic rules. A key aim is to show completeness of the proof rules
without any requirement for formal models. Establishing this for propositional
intuitionistic logic (IPL) raises some technical and conceptual issues. We
relate Sandqvist's (complete) base-extension semantics of intuitionistic
propositional logic to categorical proof theory in presheaves, reconstructing
categorically the soundness and completeness arguments, thereby demonstrating
the naturality of Sandqvist's constructions. This naturality includes
Sandqvist's treatment of disjunction that is based on its second-order or
elimination-rule presentation. These constructions embody not just validity,
but certain forms of objects of justifications. This analysis is taken a step
further by showing that from the perspective of validity, Sandqvist's semantics
can also be viewed as the natural disjunction in a category of sheaves.
