I am studying the construction of derived Quot schemes in the paper Borisov, Katzarkov, and Sheshmani - “Shifted symplectic structures on derived Quot-stacks”.

Derived quot stacks are constructed from sheaves of non-positively graded dg algebras in section 3 of the paper.

In particular, I have some question about differentials of the dg algebras.

Question

1) On the last line of page 14, a differential is constructed by the morphism $$ \mathcal{V}_j \otimes \left (\bigotimes_{1 \leq l \leq m+1} \mathcal{A}_{i_l} \right) \otimes (W_i)^\vee \rightarrow \mathcal{V}_j \otimes \left (\bigotimes_{1 \leq l \leq m} \mathcal{A}_{i_l} \right) \otimes (W_{i+i_{m+1}})^\vee. $$ However it seems to me this morphism is not a homogeneous morphism and the source and the target should be replaced by $$ \mathcal{V}_j \otimes \left (\bigotimes_{1 \leq l \leq m+1} \mathcal{A}_{i_l} \right) \otimes (W_{i+i_{m+1}})^\vee $$ and $$ \mathcal{V}_j \otimes \left (\bigotimes_{1 \leq l \leq m} \mathcal{A}_{i_l} \right) \otimes (W_{i})^\vee $$

,respectively.

Is this correct?

2) According to line 26 ~ 29 of page 14, the above differential $\delta_{\mathcal{W}}$ is constructed from the $\mathcal{A}_{\leq q-p}$-module structure on $\mathcal{W}_{[p,q]}$. But, I cannot understand the detailed definition of $\delta_\mathcal{W}$ and why this is a differential. I would appreciate it if you could tell me them.

Thank you !