This is a follow up to my previous question concerning virtual large cardinals, that are generally weaker axioms of infinity obtained from ordinary large cardinals through the so-called *virtualization* process.

If $A$ is a large cardinal property characterized by the existence of suitable embeddings. A cardinal is

virtually$A$ if the embeddings characterizing $A$ exist insome set-forcing extensions. Such a reformulation of a large cardinal axiom is called avirtualizationof $A$.

Remarkable cardinal is an instance of a virtual large cardinal axiom. By definition, it is the virtualization of a certain characterization of supercompact cardinals presented by Magidor (See ZBL0263.02034). Later Gitman and Schindler proved that the virtualization of a particular characterization of strong cardinals gives rise to an equivalent version of remarkable cardinals as well:

Theorem.The followings are equivalent:

(1)$\kappa$ is remarkable.

(2)(Virtualization of Magidor's characterization of supercompacts) For every $\lambda>\kappa$, there is $\overline{\lambda}<\kappa$ such that in a set-forcing extension there is an elementary embedding $j:V_{\overline{\lambda}}^{V}\rightarrow V_{\lambda}^{V}$ with $j(crit(j))=\kappa$.

(3)(Virtualization of a characterization of strong cardinals) For every $\lambda>\kappa$ there is $\alpha>\lambda$ and a transitive $M$ with $V_{\lambda}\subseteq M$ such that in a set forcing extension there is $j:V_{\alpha}^V\rightarrow M$ with $crit(j)=\kappa$ and $j(\kappa)>\lambda$.

Thus, virtualization doesn't preserve the consistency strength order of large cardinal axioms strictly because there are characterizations of totally different large cardinals whose virtualizations are equivalent.

Conversely, one may ask whether different characterizations of the same large cardinal axiom are always equivalent. According to what Joel mentioned in his comment here, the answer is negative! So:

Question 1.What are examples of large cardinal axioms which virtualization of their different characterizations gives rise to essentially different virtual large cardinals? Can Vopěnka's principle (which has many characterizations) be such a large cardinal axiom with non-equivalent virtual forms?

As the consistency strength of the virtualization of a certain large cardinal axiom highly depends on one's choice of its characterization, it is reasonable to ask about the possible *criteria* which guides us to choose the *right* characterization to virtualize.

Question 2.What are the criteria for choosing a certain characterization of a large cardinal to virtualize among many others?