Joel Hamkin's The set-theoretic multiverse has featured in MO questions before, e.g., here and here. But I was wondering about the best category theoretic angle to take on it.

In the paper Joel writes, rather poetically,

Set theory appears to have discovered an entire cosmos of set-theoretic universes, revealing a category-theoretic nature for the subject, in which the universes are connected by the forcing relation or by large cardinal embeddings in complex commutative diagrams, like constellations filling a dark night sky. (p. 3)

He has given us a couple of kinds of morphism here, but what is the best way to capture this multiverse category theoretically? Which morphisms should we allow?

Is it right to stay at the level of ordinary categories? Since each universe, a model of ZFC, is a category, one might expect the multiverse to be at least a bicategory, as suggested here. Do set theorists consider, say, arrows between two forcing relations between two models?