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Let $C$ be a class of structures closed under isomorphism and substructures. How can one show that there is a set $A$ of universal sentences such that the models of $A$ are precisely the elements of $C$?

Maybe the following version of the Los-Tarski theorem is helpful:

If $T$ is any first-order theory, then the models of the theory that consists of the universal sentences that hold in every model of $T$ are precisely the substructures of models of $T$.

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The claim you are asking us to prove is not true.

For example, let $C$ be the class of all finite structures in a fixed language signature. This is closed under isomorphism and substructures, but it is not the class of models of any theory, universal or otherwise, since any theory with arbitrarily large finite models must have an infinite model.

One could similarly use all countable structures or all structures of size at most $\kappa$, and make an essentially similar argument.

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