I am studying the construction of derived Quot schemes in the paper [Borisov, Katzarkov, and Sheshmani - “Shifted symplectic structures on derived Quot-stacks”][1].

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.

**Remark**

- I think $\delta_\mathcal{W}$ is defined as follows.

First, consider the morphism

$$d_{\mathcal{W}} : \mathcal{Hom} \left(\mathcal{V}_j \otimes \left (\bigotimes_{1 \leq l \leq m} \mathcal{A}_{i_l} \right), \mathcal{W}_i \right)  \rightarrow \mathcal{Hom} \left(\mathcal{V}_j \otimes \left (\bigotimes_{1 \leq l \leq m+1} \mathcal{A}_{i_l} \right), \mathcal{W}_{i+i_{m+1}} \right)$$

$$ \varphi \mapsto d_{W}(\varphi), $$

$$ d_{W}(\varphi)\left(v \otimes \bigotimes_{1 \leq l \leq m+1} a_{i_l}\right) := a_{i_{m+1}} \varphi\left(v \otimes \bigotimes_{1 \leq l \leq m} a_{i_l}\right)-a_{i_1}a_{i_2}・・a_{i_{m+1}}v.$$

And, we define $\delta_\mathcal{W}$ as its dual (but this is not a differential.)

Thank you !

  [1]: https://arxiv.org/abs/1908.03021