Check Murfet’s notes on Algebra in a category. Can the notion of a module here be used to think about representing objects for functors landing a category of representations of a group, for example the absolute Galois group???
There is the following theorem: Consider a (contravariant???) setvalued functor on a Grothendieck category. It is representable iff it commutes with small projective limits.
An interesting article by Chorny on representability for space-valued functors, also relating representability in the contravariant case to something analogous to Goodwillie calculus.
nLab page on Representability theorems