The dual crystal: reversed arrows, negated weights and swapped $\epsilon \mathrm{/}\phi$

Partial inverses and reversed arrows. By definition ${\tilde{f}}_i(b^{\vee })=b^{\mathrm{\prime }\vee }$ holds exactly when ${\tilde{e}}_{i}b=b^{\mathrm{\prime }}$, which by the partial-inverse axiom of $B$ holds exactly when $b={\tilde{f}}_i{b}^{\mathrm{\prime }}$, i.e. when $b^{\vee }={\tilde{e}}_i(b^{\mathrm{\prime }\vee })$; hence the pair $({\tilde{e}}_i,{\tilde{f}}_i)$ on $B^{\vee }$ satisfies the partial-inverse axiom. Moreover the arrows of $B^{\vee }$ are those of $B$ run backwards: an arrow $b^{\vee }\to b^{\mathrm{\prime }\vee }$ in $B^{\vee }$ means ${\tilde{f}}_i(b^{\vee })=b^{\mathrm{\prime }\vee }$, i.e. ${\tilde{e}}_{i}b=b^{\mathrm{\prime }}$, i.e. ${\tilde{f}}_i{b}^{\mathrm{\prime }}=b$, which is the arrow $b^{\mathrm{\prime }}\to b$ of $B$.

The displayed ${\epsilon }_i,{\phi }_i$ are the maximal iterated powers. By induction on $n$, ${\tilde{e}}_i^n(b^{\vee })=({\tilde{f}}_i^{n}b{)}^{\vee }$ and ${\tilde{f}}_i^n(b^{\vee })=({\tilde{e}}_i^{n}b{)}^{\vee }$; hence $\max \{n:{\tilde{e}}_i^n(b^{\vee })\mathrm{\ne }0\}=\max \{n:{\tilde{f}}_i^{n}b\mathrm{\ne }0\}={\phi }_i(b)$ and $\max \{n:{\tilde{f}}_i^n(b^{\vee })\mathrm{\ne }0\}={\epsilon }_i(b)$, as displayed; the weight pairing is preserved, ${\phi }_i(b^{\vee })-{\epsilon }_i(b^{\vee })={\epsilon }_i(b)-{\phi }_i(b)=-⟨h_i,wt(b)⟩=⟨h_i,wt(b^{\vee })⟩$, and involutivity $(B^{\vee }{)}^{\vee }=B$ holds because $(b^{\vee }{)}^{\vee }:=b$ restores the operators, weights and string data. This proves (i).

The tensor swap preserves the operators. Extend $S$ by $S(0)=0$ and put $p:={\phi }_i(b_1)$, $s:={\epsilon }_i(b_2)$. If $p\ge s$, then $S{\tilde{f}}_i((b_1\otimes b_2{)}^{\vee })=S(({\tilde{e}}_i{b}_1\otimes b_2{)}^{\vee })=b_2^{\vee }\otimes {\tilde{f}}_i(b_1^{\vee })$; the target ${\tilde{f}}_i$ selects its second factor because ${\phi }_i(b_2^{\vee })=s\le p={\epsilon }_i(b_1^{\vee })$. If $p<s$, then $S{\tilde{f}}_i((b_1\otimes b_2{)}^{\vee })={\tilde{f}}_i(b_2^{\vee })\otimes b_1^{\vee }$, and the target selects its first factor because $s>p$. For raising, if $p>s$, then $S{\tilde{e}}_i((b_1\otimes b_2{)}^{\vee })=b_2^{\vee }\otimes {\tilde{e}}_i(b_1^{\vee })$ and the target selects its second factor because $s<p$; if $p\le s$, then $S{\tilde{e}}_i((b_1\otimes b_2{)}^{\vee })={\tilde{e}}_i(b_2^{\vee })\otimes b_1^{\vee }$, and the target selects its first factor because $s\ge p$. These formulas also hold at zero endpoints. The target weight is $-wt(b_1)-wt(b_2)$, equal to the weight of the dual input; maximal string lengths are preserved because both operators are intertwined.

Conclusion. The map $S$ is a bijection of vertex sets by construction, and step 3.1 shows that it intertwines ${\tilde{e}}_i$ and ${\tilde{f}}_i$ for every $i$; it preserves weights by step 2.1, hence it is an isomorphism of crystals $(B_1\otimes B_2{)}^{\vee }\to B_2^{\vee }\otimes B_1^{\vee }$, which is (ii). Because $s$ and $p$ range over finite nonnegative integers, the case analysis covers all vertices, including the tie $p=s$ where both readings agree. All constructions are explicit set-theoretic operations on the vertex set $B$ and the fixed operator maps, so no choice principle is used; the construction is stated only for crystals of integrable-module crystal bases, the negative-half case being excluded in the Statement.