Abstract: It is an interesting task to translate properties of modules into properties of their endomorphism rings and vice versa. One classical instance of this dictionary is the characterisation of endomorphism rings of generators which goes back to work of Kato and Tachikawa. It has several more famous specialisations, for example the Morita-Tachikawa correspondence, the Auslander correspondence. We extend the generator correspondence by a tilted version which is given as an endomorphism ring in the homotopy category. We then characterise the properties of the tilt by using an intermediate extension functor corresponding