Isomorphism of a chain of structures Consider an elementary chain of models of some first-order theory $T$:
 $$ (M_\alpha)_{\alpha < \kappa}, M_\alpha \prec M_\beta  \; {\rm for} \; \alpha < \beta .$$
Let also $(N_\alpha)$ be another elementary chain of models of the same theory.
Assume, that for every $\alpha < \kappa$ ($\kappa$ a cardinal) we have an isomorphism $f_\alpha: M_\alpha \to N_\alpha$. Is there a way to extend these isomorphisms to an isomorphism
$f: M \to N$, $M:= \bigcup_{\alpha < \kappa} M_\alpha$, $N:= \bigcup_{\alpha < \kappa} N_\alpha$, even if we don't assume that the restriction of $f_\beta$ to $M_\alpha$ for $\alpha < \beta$ is equal to $f_\alpha$?
 A: No. Consider the following two linear orderings: $\mathcal{M}=\mathbb{Z}\cdot\mathbb{N}$, and $\mathcal{N}=\mathbb{Z}\cdot\mathbb{N}^*$. (Recall that $\mathcal{L}\cdot\mathcal{L}'$ is the linear order gotten by replacing each point in $\mathcal{L}'$ with a copy of $\mathcal{L}$, and that $\mathcal{L}^*$ is just $\mathcal{L}$ with the order reversed.) 
We can write $\mathcal{M}=\bigcup_{n\in\omega}\mathcal{M}_n$ and $\mathcal{N}=\bigcup_{n\in\omega}\mathcal{N}_n$, where $\mathcal{M}_n, \mathcal{N}_n\cong \mathbb{Z}\cdot n$, and such that each inclusion $\mathcal{M}_i\subseteq\mathcal{M}_j$ and $\mathcal{N}_i\subseteq\mathcal{N}_j$ ($i\le j$) is an elementary embedding.
In fact, the obstacle is precisely what you mention: that the isomorphisms between the components fail to glue together nicely.

Note that the part I'm leaving out is why the embeddings are elementary, and why $\mathcal{M}$ and $\mathcal{N}$ are elementarily equivalent; but these are straightforward arguments with Ehrenfeucht-Fraisse games.
A: Here is a simple counterexample. Let $T$ be the theory of sets with no additional structure. Let $\mathcal{M}_{\alpha}$ be the set $\omega+\alpha$ with no additional structure, and let $\mathcal{N}_{\alpha}=\mathbb{N}$ for all $\alpha<\omega_{1}$. Then if $\alpha\leq\beta$, then
$\mathcal{M}_{\alpha}$ is an elementary substructure of $\mathcal{M}_{\beta}$ and $\mathcal{N}_{\alpha}$ is an elementary substructure of $\mathcal{N}_{\beta}$. Furthermore, $\mathcal{M}_{\alpha}$ is isomorphic to $\mathcal{N}_{\alpha}$ for all countable ordinals $\alpha$. On the other hand, $\bigcup_{\alpha<\omega_{1}}\mathcal{M}_{\alpha}$ has cardinality $\aleph_{1}$ while $\bigcup_{\alpha<\omega_{1}}\mathcal{N}_{\alpha}$ is countable. The same argument holds for chains whose length is a successor cardinal. 
