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 $f\_\alpha$?



Isomorphism of a chain of structures
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.




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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.


