In some related MO questions like [The set-theoretic multiverse as a (bi)category][1] it is discussed how one might represent *the multiverse* (see [ The set-theoretic multiverse][2]) in a category theoretic way, e.g. by a category having universes as objects (models of ZFC). However it seems not so obvious which morphisms to take for such an approach (e.g. forcing relation or large cardinal embeddings or any other techniques producing new models from old ones). 

What implications do the different choices for a method to produce new set theoretic worlds from old ones have w.r.t. the multiverse / is there a *right* choice (what are the criteria)? If not, then the unfinished discussion about the right morphisms in a category theoretic representaation of a multiverse seems to just reflect this?

However, is it essential in this context to talk about *the multiverse* rather than *a multiverse*? If it is not obvious which morphisms could be best, could't one regard different multiverses or multiverse representations and treat them equally until one finds justification to single out some i.e. *the* real multiverse? 

Couldn't it be the better way to think of a multiverse as a functor $$G: T\rightarrow M$$ mapping extensions of ZFC in $T$ to ZFC-models in M, after all this seems to be natural, since a multiverse is about variable worlds (and then think about multiverses with certain properties like having a left adjoint $F:M\rightarrow T$ or to study the functor category of multiverses $M^T$ or the like)?  


  [1]: https://mathoverflow.net/questions/75836/the-set-theoretic-multiverse-as-a-bicategory
  [2]: https://arxiv.org/abs/1108.4223