Categorical separability contracts the module deformation complex

Put $\mathcal{E}={\mathrm{Fun}}_k(\mathcal{M},\mathcal{N})$ and $\mathcal{Q}={\mathrm{End}}_k(\mathcal{N})⊠{\mathrm{End}}_k(\mathcal{M}{)}^{\mathrm{r}\mathrm{e}\mathrm{v}}$. The action $(U⊠V)\cdot S=USV$ makes $\mathcal{E}$ a $\mathcal{Q}$-module category. These underlying categories are finite semisimple: after choosing their finitely many simple objects, linear functors and transformations are finite matrices of vector spaces and linear maps. The two action functors give a tensor functor $\mathcal{C}⊠{\mathcal{C}}^{\mathrm{r}\mathrm{e}\mathrm{v}}\to \mathcal{Q}$. Let $A$ be the image of $K$ and $e_A$ the image of $e$; tensor functors preserve the two separability identities. The comparisons $\gamma$ give an action ${\rho }_T:A\cdot T\to T$: on a simple summand it is the adjoint of ${\gamma }_i:N_{i}T\to T{M}_i$, namely $N_{i}T{M}_{i^{\ast }}\to T$. The product and unit equations for this action are exactly the module-functor equations, as one sees by inserting $\sum q_a^r{p}_a^r={\mathrm{i}\mathrm{d}}_{i\otimes j}$ in the displayed multiplication of $K$.

The object $B_T={\underset{}{\mathrm{End}}}_{\mathcal{Q}}(T)$ represents $W\mapsto {\mathrm{Hom}}_{\mathcal{E}}(W\cdot T,T)$: explicitly, sum each simple object of $\mathcal{Q}$ with the multiplicity of the corresponding finite-dimensional Hom space. Composition of evaluations makes $B_T$ an algebra, and ${\rho }_T$ corresponds to an algebra map $A\to B_T$. In particular $B_T$ is an $A$-bimodule. Internal-Hom adjunction and duality give isomorphisms ${\mathrm{Hom}}_{\mathcal{Q}}(A^{\otimes n},B_T)\cong {\mathrm{Hom}}_{\mathcal{E}}(A^{\otimes n}\cdot T,T)\cong C_T^n\mathrm{.}$ For the last map, the summand indexed by $(i_1,\dots ,i_n)$ is the adjunction ${\mathrm{Hom}}_{\mathcal{E}}(N_{i_1\otimes \cdots \otimes i_n}T{M}_{(i_1\otimes \cdots \otimes i_n{)}^{\ast }},T)\cong {\mathrm{Hom}}_{\mathcal{E}}(N_{i_1\otimes \cdots \otimes i_n}T,T{M}_{i_1\otimes \cdots \otimes i_n})\mathrm{.}$ The module associators identify tensor-word actions with the iterated actions in the statement. Values on simple tuples extend uniquely to natural transformations because every object is a finite direct sum of simples. Thus these maps are isomorphisms, not just injections into the cochain spaces.

Under these isomorphisms the two outer Hochschild faces compose with ${\rho }_T$ on the two sides; taking their adjoints gives precisely the zeroth and last cofaces displayed in the statement. For an interior face, the component ${\sum }_a{p}_a^r⊠(q_a^r{)}^{\ast }$ of multiplication inserts ${\sum }_{r,a}{q}_a^r{p}_a^r=\mathrm{i}\mathrm{d}$ in the adjacent tensor factors; its adjoint is the merge coface, transported by the module associators. The unit of $A$ is the image of the unit summand of $K$, so the normalized conditions also agree. Consequently the maps in the preceding step identify the stated normalized complex with the normalized Hochschild complex of $A$ with coefficients $B_T$. This also proves $d^2=0$ for the stated differential.

Denote this complex isomorphism by ${\mathrm{\Phi }}_n:C_T^n\to {\mathrm{Hom}}_{\mathcal{Q}}(A^{\otimes n},B_T)$. For $n\ge 1$ define the required homotopy by $h_n(f)={\mathrm{\Phi }}_{n-1}^{-1}({\lambda }_{B_T}({\mathrm{i}\mathrm{d}}_A\otimes {\mathrm{\Phi }}_n(f))(e_A\otimes {\mathrm{i}\mathrm{d}}_{A^{\otimes (n-1)}}))\mathrm{.}$ All maps in this formula have been specified in the preceding steps. Fact [F2] and the complex isomorphism give $d{h}_n+h_{n+1}d={\mathrm{i}\mathrm{d}}_{C_T^n}$ and preservation of normalization. The same proof with $T={\mathrm{i}\mathrm{d}}_{\mathcal{M}}$ gives the action complex; replacing $\mathcal{C}$ by its tensor reversal gives the prescribed right-action convention. Thus every positive-degree cocycle is a coboundary, in particular every degree-two cocycle.