A class of $L$ structures $S$ has the amalgamation property if and only if whenever $A,B_{1},B_{2} \in S$ and $f_{i}:A \ra B_{i}$ are elementary embeddings for $i \in \{1,2\}$ then there is some $C \in S$ and some elementary embeddings $g_{i}:B_{i} \ra C$ 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 and the definition of pushout contained there.