The problem is not whether you can use Yoneda lemma or not, but to associate a precise meaning to your sentences. As a rule of thumb, Yoneda lemma works whenever you can formally say what it actually means.

I do not agree with Zhen's answer which says that it is not really important that the objects are not inside the universe --- the truth is exactly the opposite --- it is the only issue here. In the usual ZFC formalization, elements of $FA$ are real sets, whereas elements of $\mathit{nat}(h_A, F)$ are not. Of course you can "lift" $FA$ to the meta-universe, but then you will compare apples with apes.

The problem is not whether you can use Yoneda lemma or not, but to associate a precise meaning to your sentences. As a rule of thumb, Yoneda lemma works whenever you can formally say what it actually means.

I do not agree with Zhen's answer which says that it is not really important that the objects are not inside the universe --- the truth is exactly the opposite --- it is the only issue here. In the usual ZFC formalization, objects $FA$ are real sets, whereas objects $\mathit{nat}(h_A, F)$ are not. Of course you can "lift" $FA$ to the meta-universe, but then you will compare apples with apes.