Tensor and direct extension of definable subcategories

Definable subcategories may be extended along a ring homomorphism directly, by using their defining conditions in the new module category, or by tensoring up with the new ring. We investigate what is preserved and reflected by these processes. On the way, we introduce the notion of being Mittag-Leffler with respect to a bimodule - a refinement of the Mittag-Leffler condition. Particular attention is given to the case where the ring homomorphism is an elementary embedding.