Dual-fusion Hom and conformal-block identification

Since A' is a module by F2, F1 applies with $Z=A^{\mathrm{\prime }}$. It identifies each block $\mathrm{\Gamma }\in {\mathcal{T}}_G^{\ast }(X\otimes A^{\mathrm{\prime }})$ with a unique module map $f:A^{\mathrm{\prime }}\to B$, characterized by $\mathrm{\Gamma }(x\otimes a^{\mathrm{\prime }})={\gamma }_G(x\otimes f(a^{\mathrm{\prime }}))$. The incoming factor X remains first; no tensor permutation or braiding is introduced.

A module map f commutes with every component $L_j(0)$, so it preserves joint generalized weights. Its restricted transpose $f^{\mathrm{\prime }}:B^{\mathrm{\prime }}\to A^{\mathrm{\prime }\mathrm{\prime }}$ is therefore well-defined. For any component state u and formal variable z, pairing with $a^{\mathrm{\prime }}\in A^{\mathrm{\prime }}$ gives $⟨f^{\mathrm{\prime }}{Y}_{B^{\mathrm{\prime }}}(u,z)b^{\mathrm{\prime }},a^{\mathrm{\prime }}⟩=⟨b^{\mathrm{\prime }},Y_B^o(u,z)f(a^{\mathrm{\prime }})⟩=⟨b^{\mathrm{\prime }},f(Y_{A^{\mathrm{\prime }}}^o(u,z)a^{\mathrm{\prime }})⟩=⟨Y_{A^{\mathrm{\prime }\mathrm{\prime }}}(u,z)f^{\mathrm{\prime }}(b^{\mathrm{\prime }}),a^{\mathrm{\prime }}⟩$. The middle equality follows from f commuting with every original mode and therefore with its finite opposite-state combinations. The separating pairing proves that f' is a module map. Consequently $g=e_A^{-1}{f}^{\mathrm{\prime }}:B^{\mathrm{\prime }}\to A$ is a module map by F3.

This correspondence is exactly the transpose-pairing identity $⟨a^{\mathrm{\prime }},g(b^{\mathrm{\prime }})⟩=⟨b^{\mathrm{\prime }},f(a^{\mathrm{\prime }})⟩$. Conversely, for any module map $g:B^{\mathrm{\prime }}\to A$, the same transposition argument gives the module map $f=e_B^{-1}{g}^{\mathrm{\prime }}:A^{\mathrm{\prime }}\to B$ and that identity. It determines f and g uniquely: restricted functionals separate each finite joint block, and both maps preserve those blocks. Thus the two constructions are inverse linear bijections. Composing this bijection with step 1.1 gives precisely the isomorphism in the Statement, without finite-dimensionality of either entire module or either Hom space.

Naturality follows from the same two explicit identities. The representing equality in step 1.1 is natural by F1. The transpose-pairing identity commutes with precomposition and postcomposition by module maps: moving such a map across its evaluation pairing replaces it by its restricted transpose, by definition. F3 ensures that the evaluation identifications used in this movement are natural. Hence the composed linear bijection is natural in the module data and respects the specified incoming/outgoing ordering. No semisimplicity or rationality is used.