Approximations of groups, subquotients of infinite direct products and equations over groups

Let C be a class of groups. (For example, C is a class of all finite groups, or C is a class of all finite symmetric groups.) I give a definition of approximations of a group G by groups from C. For example, the groups approximable by symmetric groups are, by definition, sofic groups.

For some classes C the following result holds:

G is approximable by C if and only if G is a subgroup of some quotient of a (infinite) direct product of groups from C.

It also may  be formulated using equations over groups. In the talk I plan to explain this in details and discuss some related results and open questions.