As suggested, let $|\mathfrak{B}|=\mbox{dom}h$ , and we define the structure $\mathfrak{B}$ having the universe $|\mathfrak{B}|$ . Note that $h:|\mathfrak{B}|\rightarrow|\mathfrak{A}|$ and is onto. For any $b\in|\mathfrak{A}|$ , let $g(b)$ be any point from the non-empty set $G(b)=\{a\in|\mathfrak{B}||h(a)=b\}$ (here we use the Axiom of Choice). For any predicate symbol $P$ , let $<a_{1},\ldots,a_{n}>\in P^{\mathfrak{B}}$ iff $<h(a_{1}),\ldots,h(a_{n})>\in P^{\mathfrak{A}}$ . For any constant symbol $c$ , let $c^{\mathfrak{B}}=g(c^{\mathfrak{A}})$ . Similarly, for any function symbol $f$ , let $f^{\mathfrak{B}}(a_{1},\ldots,a_{n})=g(f^{\mathfrak{A}}(h(a_{1}),\ldots,h(a_{n})))$ . Then, the homomorphism property holds for any predicate parameter by definition. For any constant parameter, $h(c^{\mathfrak{B}})=h(g(c^{\mathfrak{A}}))\in h(G(c^{\mathfrak{A}}))=\{c^{\mathfrak{A}}\}$ . And for every function parameter, $h(f^{\mathfrak{B}}(a_{1},\ldots,a_{n}))=h(g(f^{\mathfrak{A}}(h(a_{1}),\ldots,h(a_{n}))))\in h(G(f^{\mathfrak{A}}(h(a_{1}),\ldots,h(a_{n}))))=\{f^{\mathfrak{A}}(h(a_{1}),\ldots,h(a_{n}))\}$ . Therefore, $h$ is a homomorphism of $\mathfrak{B}$ onto $\mathfrak{A}$ .