We investigate the problem of punctual (fully primitive recursive) presentability of algebraic structures up to primitive recursive and computable isomorphism. We show that for mono-unary structures and undirected graphs, if a structure is not punctually categorical then it has infinitely many punctually non-isomorphic punctual presentations. We also show that the punctual degrees of any computably almost rigid structure as well as the order ( $$\mathbb {Z},<$$ Z, < ) are dense. Finally we characterise the Boolean algebras which have a punctually (...) 1-decidable presentation that is computably isomorphic to a 1-decidable presentation. (shrink)