Local channel projections are orthogonal idempotents

Let $b,c,a_1,\dots ,a_n$ be simple objects of a unitary fusion category, with $n\ge 2$, and let $T_{\mathrm{f}\mathrm{i}\mathrm{x}\mathrm{e}\mathrm{d}}$ be a fixed parenthesization of $b\otimes a_1\otimes \cdots \otimes a_n$. Equip $H=\mathrm{Hom}(c,T_{\mathrm{f}\mathrm{i}\mathrm{x}\mathrm{e}\mathrm{d}})$ with the split Hom inner product $g^{†}h=⟨h,g⟩{\mathrm{id}}_c$ and choose orthonormal split vertex bases in the trees below. Fix $1\le i<n$. Let $A_i:T_{\mathrm{f}\mathrm{i}\mathrm{x}\mathrm{e}\mathrm{d}}\to T_{\mathrm{p}\mathrm{a}\mathrm{i}\mathrm{r}}$ be a composite of associators and inverses to a tree fusing $a_i,a_{i+1}$ first, without permuting inputs. Let $F_i$ be the matrix of $h\mapsto A_i\circ h$, with target rows and source columns, and let ${\mathrm{\Pi }}_s$ project onto all pair-first coordinates of simple pair output $s$, retaining every multiplicity index. Define $P_{i,s}:=F_i^{-1}{\mathrm{\Pi }}_s{F}_i$, extending the repeated-charge construction of Local anyonic channel projector to these arbitrary simple site charges. Then $(P_{i,s}{)}_s$ is pairwise orthogonal, self-adjoint, and idempotent, and its sum over all simple outputs of $a_i\otimes a_{i+1}$ is the identity on $H$.