Yes, the relation is that they anti-commute.
Let's see this very briefly (you can find it in almost all books on Kähler geometry and Hodge theory).
We want to compute [D″E,delta′E], where [bullet,bullet] is the graded commutator and let me call omega (instead of g) the Kähler form. Then one has that
[Lambdaomega,D″E]=−idelta′E,
so that
[D″E,delta′E]=i[D″E,[Lambdaomega,D″E]],
where Lambdaomega=∗−1Lomega∗ is the formal adjoint of the operator of degree (1,1) given by Lomegabullet=omegawedgebullet.
Now, the (graded) Jacobi identity gives
−[D″E,[Lambdaomega,D″E]]+[D″E,underbrace[D″E,Lambdaomega]=−[Lambdaomega,D″E]]+[Lambdaomega,underbrace[D″E,D″E]=0]=0,
thus −2[D″E,[Lambdaomega,D″E]]=0 and
D″Edelta′E+delta′ED″E=[D″E,delta′E]=0.
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