linguistic:paraphrasing
Upward part

First, one extends the signature by adding a new constant symbol for every element of M {\displaystyle M}. The complete theory of M {\displaystyle M} for the extended signature σ ′ {\displaystyle \sigma '} is called the elementary diagram of M {\displaystyle M}. In the next step one adds κ {\displaystyle \kappa } many new constant symbols to the signature and adds to the elementary diagram of M {\displaystyle M} the sentences c ≠ c ′ {\displaystyle c\neq c'} for any two distinct new constant symbols c {\displaystyle c} and c ′ {\displaystyle c'}. Using the compactness theorem, the resulting theory is easily seen to be consistent. Since its models must have cardinality at least κ {\displaystyle \kappa }, the downward part of this theorem guarantees the existence of a model N {\displaystyle N} which has cardinality exactly κ {\displaystyle \kappa }. It contains an isomorphic copy of M {\displaystyle M} as an elementary substructure.