Now let us consider an arbitrary tuple $a=(S_1,\dots,S_n)$ of ${\mathsf{UnionSing}}({\wp}({\mathbf{B}})^{n})$ and an arbitrary $b \in s(a)$. Since $\bigcup_{i \in [n]} S_i = \bigcup_{i \in [n], |S_i| = 1} S_i$, there is an $i\in[n]$ such that $\{b\}=S_i$. Thus, the homomorphism $\pi_i\colon {\mathsf{UnionSing}}({\wp}({\mathbf{B}})^{n}) \rightarrow {\wp}({\mathbf{B}})$ that projects onto the $i$th coordinate satisfies $\pi_i(a)=\{b\}$, and for all tuples $a'$ of ${\mathsf{UnionSing}}({\wp}({\mathbf{B}})^{n})$, it holds that $\pi_i(a')\subseteq s(a')$. Hence, $({\mathsf{UnionSing}}({\wp}({\mathbf{B}})^{n}),{\mathbf{B}})$ has the SACC.

For the other direction, we show that if there is a homomorphism from ${\mathsf{UnionSing}}({\wp}({\mathbf{B}})^{n})$ to ${\mathbf{B}}$ for all $n \geq 1$, then singleton arc consistency solves ${{\mathsf{CSP}}}({\mathbf{B}})$. Thus, we have to show that there exists a homomorphism from ${\mathbf{A}}$ to ${\mathbf{B}}$ if $({\mathbf{A}},{\mathbf{B}})$ has the SACC. Let $s$ be the homomorphism from the definition of SACC, and let us use $\{h_1,\ldots ,h_n\}$ to denote the set ${\{h_{a,b}\mid a\in A,b\in s(a)\}}$ of homomorphisms. Further, let $g$ be the homomorphism $(h_1,\ldots ,h_n) \colon {\mathbf{A}}\to {\wp}({\mathbf{B}})^{n}$. Now, for every element $a\in A$ the image $g(a)=(h_1(a),\ldots ,h_n(a))$ is a tuple of ${\mathsf{UnionSing}}({\wp}({\mathbf{B}})^{n})$: for every $b\in \bigcup_{j \in [n]} h_j(a)$, it holds that $b \in s(a)$ and thus there exists a homomorphism $h_{a,b}=h_i$ that maps $a$ to the singleton $\{b\}$; so, we have $\bigcup_{j \in [n]} h_j(a) 