I have come across two different definitions of saturation and I would like to relate them.
One is in Wilfrid Hodges book (first snippet below) and it says that an elementary extension $B$
realizes all types with less than $<\lambda$ elements. The other approach is the definition 1.0.25 in the second snippet below.
It says that $M$ of cardinality $\lambda^+$ is saturated if $M$ realizes type of every sub-model of cardinality $\lambda$. I do not even know what is to be proved for an equivalence of these two
approaches. It appears to me that in the second approach we are looking at **smaller** models and in the first approach at **larger** elementary extensions.



[![enter image description here][1]][1]

[![enter image description here][2]][2]


  [1]: https://i.sstatic.net/Hy5zN.jpg
  [2]: https://i.sstatic.net/eAkhm.jpg