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?

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 abelian groups and commutative ring objects in some category $\mathsf C$ depending on the completeness and exactness properties of $\mathsf C$.

I am also interested in the behavior of objects and arrows defined by lifting properties, e.g injectives, strong epis, etc. For instance, if $\mathsf C$ has enough injectives, when does this hold for (simplicial) algebraic objects in $\mathsf C$?