A structure is locally finite if every finitely generated substructure is finite; local sentences are universal sentences all models of which are locally finite. The stretching theorem for local sentences expresses a remarkable reflection phenomenon between the finite and the infinite models of local sentences. This result in part requires strong axioms to be proved; it was studied by the second named author [J.S.L. 53, No. 4, p. 1009-1026]. Here we correct and extend this paper; in particular we show that the stretching theorem implies the existence of inaccessible cardinals, and has precisely the consistency strength of Mahlo cardinals of finite order. And we present a sequel due to the first named author: (i) decidability of the spectrum Sp( phi ) of a local sentence phi, below omega^omega ; where Sp( phi ) is the set of ordinals alpha such that phi has a model of order type alpha; (ii) proof that beth_omega =sup { Sp( phi ) : phi local sentence with a bounded spectrum}; (iii) existence of a local sentence phi such that Sp( phi ) contains all infinite ordinals except the inaccessible cardinals.