If we have a family of classes $(\mathfrak{M}\_\alpha)\_{\alpha\in D}$ of $\in$-structures with $D$ being a limit ordinal or the class of ordinals, and a family $(\chi\_{\alpha,\beta})\_{\alpha<\beta(<D)}$ being a direct system (see [here][1] for a definition of a direct system) of end extensions (an end extension $\chi:\langle A,E\rangle\rightarrow\langle B,F\rangle$ being a 1-1 $\in$-homomorphism such that $bF\chi(a)\Leftrightarrow b=\chi(a')$ for some $a'Ea$), how can one define the direct limit of the $(\mathfrak{M}\_\alpha,\chi\_{\alpha,\beta})$ so that the result is still a class ?


I suppose that the $\chi\_{\alpha,\beta}$ are not necessarily compatible, in the sense that given $\alpha$, $\chi\_{\alpha,\beta}(x)$ is not necessarily equal to $\chi\_{\alpha,\beta'}(x)$ for $\beta\neq\beta'$.



How can one define the direct limit of classes?
You can define it the same way as you define a direct limit of sets, using [Scott's trick](http://en.wikipedia.org/wiki/Scott%27s_trick) to form equivalence classes. Whenever you have an equivalence relation defined on a class $X$, you can form equivalence classes that are sets by sending $x\in X$ to the set of all $y\in X$ that are equivalent to $x$ and have minimal rank (in the sense of the cumulative hierarchy).


