= Solution
The <Chern connection> decomposes on <vector-bundle-valued differential forms> as $D=D'+D''$, where $D'=\nabla'_h$ raises holomorphic degree and $D''=\nabla''_h=\bar\partial_E$ raises antiholomorphic degree. In a <holomorphic local frame>, $D'=\partial+A\wedge$ and $D''=\bar\partial$. The type of the <curvature form of a connection> gives
$$
(D')^2=(D'')^2=0,\qquad D'D''+D''D'=F_h\wedge.
$$
The <Kähler metric> and the <Hermitian metric> define the $L^2$ inner product and the <formal adjoints> $D'^*,D''^*$. The <Dolbeault Laplacians> are
$$
\Delta'=D'D'^*+D'^*D',\qquad
\Delta''=D''D''^*+D''^*D''.
$$
Let $L=\omega\wedge$ be the <Lefschetz operator of a Kähler manifold> and $\Lambda=L^*$ its <adjoint Lefschetz operator>. With the ordinary commutator convention $[A,B]=AB-BA$, the printed <Kähler identities> give
$$
D'^*=i[\Lambda,D''],\qquad D''^*=-i[\Lambda,D'].
$$
Substitution into the two <Dolbeault Laplacians>, followed by expansion, yields
$$
\begin{aligned}
\Delta''-\Delta'
&=-i\{D'', [\Lambda,D']\}-i\{D',[\Lambda,D'']\}\\
&=i[D'D''+D''D',\Lambda]\\
&=[iF_h\wedge,\Lambda].
\end{aligned}
$$
The middle line follows by cancelling the terms with $D'\Lambda D''$ and $D''\Lambda D'$; the remaining terms collect the anticommutator of the two differentials. Hence \b[$\boxed{\Delta''=\Delta'+[iF_h,\Lambda]}$], where $iF_h$ denotes its wedge action. This is the <Bochner-Kodaira-Nakano identity> with the paper's sign convention.
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