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 this should be replaced by the morphism $$ \mathcal{V}_j \otimes \left (\bigotimes_{1 \leq l \leq m+1} \mathcal{A}_{i_l} \right) \otimes (W_{i+i_{m+1}})^\vee \rightarrow \mathcal{V}_j \otimes \left (\bigotimes_{1 \leq l \leq m} \mathcal{A}_{i_l} \right) \otimes (W_{i})^\vee. $$ Is this correct?

2) Does the differential $\delta_W$ constructed from the above morphism really become a differential? It seems to me that ${\delta_W}^2 \neq 0$. I would appreciate it if you could tell me the detailed construction of the differential (including sign convention).

3) Do we need the differential on $C^{\bullet}$ (not on $\mathcal{C}^{\bullet}$) constructed from the multiplication on $\oplus \mathcal{A_i}$ like that on $B^{\bullet}$ on line 10 of page 13?

Thank you !