A concrete category is a pair $(C,U)$ where $C$ is a category and $U$ is a faithful functor $C \to Set$. An abelian category is an additive category in which every morphism admits a kernel, a cokernel and its respective parallel morphism is iso.
I'm interested in generalizing the concept of free resolutions for abelian categories which are also concretizable. Which kind of the coherence condition between the concretization functor and the abelian structure is the most "natural"?
Slightly related question: Definition of a *Monoidal Abelian Category*