You are right, I should have said it is a "commutative diagram" of models.
Hamkins considers ill-founded models, which I guess is not super popular, plus he considers not just "M is a transitive subclass of N" but all kinds of embeddings, like those coming from ultrapowers, so it is indeed more...
Let ##(M_i)_{i\in I}## be a multiverse of models of ZFC. By that I mean:
Each ##M_i## is a well-founded model of ZFC.
##(I,\leq_I)## is a partially ordered set, and whenever ##i\leq_I j##, there is an embedding ##\tau^j_i:M_i\rightarrow M_j## such that the image of ##M_i## is a transitive...