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This is where you make an easily made error (emphasis mine)

how can it be that the category Set is complete under small limits?

Set depends on your background choice of set theory (if that is how you are doing things). Your question has been answered, but I hope this will clear up a few loose ends. Given two models $M_1$, $M_2$ of ZF(C), you get two categories $Set_1$ and $Set_2$. Now both of these are cartesian closed, so that the hom-objects of $Set_i$ are objects of $Set_i$.

• Point 1: nothing tells us that the hom-objects of one are hom-objects objects of the other.

Now both of these have limits, with respect to diagrams $D_i \to Set_i$ _where $D_i$ (the shape of the diagram) is a category internal to $Set_i$_.

• Point 2: nothing tells us that $D_1$ is a category internal to $Set_2$ and vice versa.

But I gather from your question that you are interested in diagrams of shape $D$ such that $D$ is a category in both $Set_1$ and $Set_2$. (This poses somewhat of a restriction, especially if $M_1$ (say) is a countable model, as then $D_1$ has a countable set of arrows, whereas $Set_2$ may have many more limits) Now suppose we have such a $D$. Then all diagrams $D\to Set_i$ have a limit. This argument is rubbery, as it supposes that $D$ is contained in two categories at once, so see below.

However, your question is, what happens if these limits are different? Well, this is not really a question one can ask, as $Set_1$ and $Set_2$ are different categories. There is no a priori way to compare objects of two different categories. If one has a functor between categories, then it is possible to see what relation objects of one category have to another category (at this point, someone might ask 'what about comparing the objects of the two categories in some meta-theory, but taking a category theoretic approach, one does not refer to the meta-theory - everything happens inside the categories). But to have such a functor one needs $Set_1$ and $Set_2$ to belong to the same category $Cat$ of categories. But $Cat$ again depends on one's choice of background theory. This is not a problem, because it makes less sense to ask for a morphism between two objects of a different category than to ask to compare objects of different categories. So in a sense, $Set_1$ and $Set_2$ don't know about each others' limits.

So really where things 'went wrong' is that we assumed that $D$ was contained in both categories $Set_1$ and $Set_2$. To even know this works we need a functor between $Set_1$ and $Set_2$. But what $D$ is in $Set_1$ could be somewhat different to what it is in $Set_2$. We could take $D = \mathcal{C}$ from Andrej's answer, which is not a single category but really two, $D_1$, $D_2$ one in each of $Set_i$. The functor between the two categories of sets can quite happily map one of these categories to the other, and the limit will be preserved by this functor, it's just they won't be "the same".

The above is very much off-the-cuff, and I am not a logician, but I hope it helps a little.

1

This is where you make an easily made error (emphasis mine)

how can it be that the category Set is complete under small limits?

Set depends on your background choice of set theory (if that is how you are doing things). Your question has been answered, but I hope this will clear up a few loose ends. Given two models $M_1$, $M_2$ of ZF(C), you get two categories $Set_1$ and $Set_2$. Now both of these are cartesian closed, so that the hom-objects of $Set_i$ are objects of $Set_i$.

• Point 1: nothing tells us that the hom-objects of one are hom-objects of the other.

Now both of these have limits, with respect to diagrams $D_i \to Set_i$ _where $D_i$ (the shape of the diagram) is a category internal to $Set_i$_.

• Point 2: nothing tells us that $D_1$ is a category internal to $Set_2$ and vice versa.

But I gather from your question that you are interested in diagrams of shape $D$ such that $D$ is a category in both $Set_1$ and $Set_2$. (This poses somewhat of a restriction, especially if $M_1$ (say) is a countable model, as then $D_1$ has a countable set of arrows, whereas $Set_2$ may have many more limits) Now suppose we have such a $D$. Then all diagrams $D\to Set_i$ have a limit. This argument is rubbery, as it supposes that $D$ is contained in two categories at once, so see below.

However, your question is, what happens if these limits are different? Well, this is not really a question one can ask, as $Set_1$ and $Set_2$ are different categories. There is no a priori way to compare objects of two different categories. If one has a functor between categories, then it is possible to see what relation objects of one category have to another category (at this point, someone might ask 'what about comparing the objects of the two categories in some meta-theory, but taking a category theoretic approach, one does not refer to the meta-theory - everything happens inside the categories). But to have such a functor one needs $Set_1$ and $Set_2$ to belong to the same category $Cat$ of categories. But $Cat$ again depends on one's choice of background theory. This is not a problem, because it makes less sense to ask for a morphism between two objects of a different category than to ask to compare objects of different categories. So in a sense, $Set_1$ and $Set_2$ don't know about each others' limits.

So really where things 'went wrong' is that we assumed that $D$ was contained in both categories $Set_1$ and $Set_2$. To even know this works we need a functor between $Set_1$ and $Set_2$. But what $D$ is in $Set_1$ could be somewhat different to what it is in $Set_2$. We could take $D = \mathcal{C}$ from Andrej's answer, which is not a single category but really two, $D_1$, $D_2$ one in each of $Set_i$. The functor between the two categories of sets can quite happily map one of these categories to the other, and the limit will be preserved by this functor, it's just they won't be "the same".

The above is very much off-the-cuff, and I am not a logician, but I hope it helps a little.