How to extend structure $A$ so said structure is isomorphic to $B$?

Question

Let $A,B$ be structures of the same type such that we have an embedding $\phi:A\longrightarrow B$ (which is strong by definition). How can we prove there is a structure $C$ of the same type such that $A\subseteq C$ and there is a $g:C\longrightarrow B$ an isomoprhism?

Answer 1

Hint: We can define a structure isomorphic to $B$, on $C:=A\sqcup (B\setminus\phi(A))$ where $\sqcup$ denotes disjoint union.