I have a question about the following paper.

**ONE-DIMENSIONAL ASYMPTOTIC CLASSES OF FINITE STRUCTURES**

By: Macpherson and Steinhorn

[https://www.ams.org/journals/tran/2008-360-01/S0002-9947-07-04382-6/S0002-9947-07-04382-6.pdf][1]

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Let $\mathbf{K}$ be a one-dimensional asymptotic class of finite structures.


The phrase "**every infinite ultraproduct**" appear in this paper several times. For example Lemma 2.5.  This phrase is vague for me. What does that mean? Does it mean that the ultraproduct of evey infinite subclass of $\mathbf{K}$? or it means the ultraproduct of **all** members of  $\mathbf{K}$ under different ultrafilters? 


  [1]: https://www.ams.org/journals/tran/2008-360-01/S0002-9947-07-04382-6/S0002-9947-07-04382-6.pdf