Given a category, one is often interested in the category of (abelian) group and (commutative) ring objects in it. I would like to know what exactness properties such categories and their simplicial analogues have, e.g what can be said about the simplicial commutative ring objects in some category $\mathsf C$ depending on the completeness and exactness properties of $\mathsf C$.

As a naive question which probably has a negative answer: is there a functorial assignment of a category of specified algebraic objects in a given category $\mathsf C$?

What are some other approaches?