Does the class category of ZF-algebras satisfy the Multiverse axioms?

My understanding of algebraic set theory is imperfect, but I am given to understand that what is involved is some kind of category consisting of class models of set theory. To formalize it, I would want to ask the question in the context of a background set theory, such as GBC or some other second-order set theory, where one has a universe $V$ of ZFC, and the corresponding family of classes of $V$, which satisfy the GBC axioms of set theory. Let me assume that the intended category is included within the family of these classes that are models of some fragment of ZFC, that is, classes of the form $(W,E)$, where $W\subset V$ is a class of objects and $E\subset W\times W$ is a binary relation on $W$ satisfying the ZFC axioms. And furthermore, allow me to assume that $(V,\in)$ itself is supposed to be one of the models in the intended category.

If this understanding of the situation is correct (and please correct me if it is not), then there will be problems with several of the multiverse axioms, and they will not all hold in this setting. Let me mention a few of the problems.

First, the forcing extension axiom will fail (see my paper The set-theoretic multiverse for the axioms). This is the multiverse axiom asserting that if $W$ is any universe and $\mathbb{P}\in W$ is a forcing notion in $W$, then there will be a corresponding $W$-generic forcing extension $W[G]$ with $G\subset\mathbb{P}$. If $V$ itself is one of the universes, however, then there will have to be classes in $V$ that represent forcing extensions $V[G]$ by nontrivial forcing notions $G\subset\mathbb{P}\in V$. But this is impossible, since no such generic filters $G$ can exist in $V$, and therefore no such class can exist in $V$. (There is a more subtle sense in which one can have such classes, if one uses the Boolean ultrapower understanding of forcing, by which one has an elementary embedding $j:V\to\overline{V}$ such that in $V$ there is a $\overline{V}$-generic filter $G\subset j(\mathbb{P})$, but this is not quite the same thing.)

The previous argument assumed that $V$ was one of the universes in the intended category. One could weaken this just to the assertion that there was some universe $W$ in the category that had the true $\omega$ and also some uncountable sets in it. In this case, the forcing to collapse those sets to $\omega$ would be a forcing notion in $W$, but could have no $W$-generic filter in $V$, since any such filter would really collapse those sets to become countable, which they are not in $V$, and so there would be none of the required forcing extensions of $W$ in the category. Basically, the forcing extension axiom asserts that one must really reach outside the current set-theoretic background to achieve it, and so if one builds the category of universes inside the set-theoretic background, one will not have the generic objects available.

Similar reasoning shows that many of the other multiverse axioms will fail for this category. For example, some of the multiverse axioms imply that every universe is a countable transitive model inside another universe of the multiverse. But again, if the category of universes includes any actually uncountable transitive universes, then this is impossible inside any class model of $V$.

Further problems arise for such a category of universes with the well-founded mirage axiom, which asserts that every universe is seen as having a non-standard $\omega$ by another universe in the multiverse (one of the more controversial axioms). This axiom will fail if the category of universes is to include any model with what is to the $V$ perspective the standard $\omega$, since $V$ can have no class model that looks upon the true $\omega$ of $V$ as ill-founded, since then $V$ itself would look upon its own $\omega$ as ill-founded.

Meanwhile, in my paper A natural model of the multiverse axioms with Victoria Gitman, we prove that if ZFC is consistent, then the collection of computably-saturated countable models of ZFC is a model of all the multiverse axioms (and furthermore any collection of models of set theory satisfying the multiverse axioms must consist entirely of computably saturated models). Many of our arguments in that paper involve the kind of algebraic treatment of models of set theory that I naively expect to be a part of the analysis in algebraic set theory, and so my expectation is that if there are bridges to be built or discovered between the two perspectives that you mention, I would expect to find them there, in the realm of collections of highly saturated models of set theory.

I think the two theories should be regarderd as existing at different levels. A category of classes, in the sense of algebraic set theory, is not a collection of models of set theory, but (an abstraction of) the collection of all classes relative to one model of set theory. In particular, if $V$ is a model of set theory, then the collection of all classes in V is a category of classes.

One can then, if one wants, define a notion of "internal model of set theory" inside a category of classes, and in the "canonical" example this would just be the notion of a class-model of set theory relative to $V$. The model $V$ itself is recoverable as the "initial" such internal model. Joel's excellent answer explains why such a collection of "internal models" will not generally satisfy the multiverse axioms.