Let $(X,g)$ be an Hermitian manifold, and $(E,h)$ be an Hermitian vector bundle over $X$, equipped with an integrable, unitary connection $D=D'+D''$. Let $\beta\in\Lambda^{p,q}(\mathrm{End}\,E)$ be unitary. Then it is well-known that the induced connection on $\mathrm{End}\,E$ satisfies the identity \begin{equation} D'_{\mathrm{End}}(\beta)=D'\circ\beta-(-1)^{p+q}\beta\circ D'\in\Lambda^{p+1,q}(\mathrm{End}\,E) \end{equation} Consider now the formally adjoint operator $(D'')^*=-*D'*:\Lambda^{p,q}(E)\rightarrow\Lambda^{p,q-1}(E)$. Note that $D''_{\mathrm{End}}$ also has a formally adjoint operator $(D''_{\mathrm{End}})^*=-*D'_{\mathrm{End}}*:\Lambda^{p,q}(\mathrm{End}\,E)\rightarrow\Lambda^{p,q-1}(\mathrm{End}\,E)$.
Question: Is there a similar identity for $(D''_{\mathrm{End}})^*(\beta)$ in terms of $(D'')^*$ and $\beta$?
