amalgamation property

A class of $L$ -structures $S$ has the amalgamation property if and only ifwhenever $A,B_{1},B_{2}\\in S$ and $f_{i}:A\\rightarrow B_{i}$ are elementary embeddingsfor $i\\in\\{1,2\\}$ then there is some $C\\in S$ and some elementary embeddings $g_{i}:B_{i}\\rightarrow C$ for $i\\in\\{1,2\\}$ so that $g_{1}(f_{1}(x))=g_{2}(f_{2}(x))$ for all $x\\in A$ . That is, the following diagram commutes.

\\xymatrix{&{A}\\ar[dl]_{f_{1}}\\ar[dr]^{f_{2}}&\\\\{B_{1}}\\ar[dr]_{g_{1}}&&{B_{2}}\\ar[dl]^{g_{2}}\\\\&{C}&}

Compare this with the free product with amalgamated subgroup for groups andthe definition of pushout there.