I am reading the paper "Revisiting the de Rham-Witt complex" by Bhatt-Lurie-Mathew and I am a bit confused about the difference between $W\Omega_R^*$ and $\hat{\Omega}^*_{W(R)}$.
Let $\mathrm{CAlg}_{\mathrm{\mathbb{F}_p}}^{\mathrm{reg}}$ be the category of regular noetherian $\mathbb{F}_p$-algebras. In my understanding, the motivation to define the de Rham-Witt complex is to construct a functor $$ \mathrm{CAlg}_{\mathrm{\mathbb{F}_p}}^{\mathrm{reg}}\to \mathrm{Ch}(\mathrm{Ab}) $$ which can be described explicitly and computes the crystalline cohomology. To construct such a functor, they are introducing the notion of "strict Dieudonne complex" and defining $W\Omega_R^*$ to be the "strictification" of $\hat{\Omega}^*_{W(R)}$. However, it seems to me that the functor $\hat{\Omega}^*_{W(-)}$ already has the required property. Indeed, their proof of the comparison theorem (between de Rham-Witt and crystalline cohomology) relies only on the following theorem:
Theorem 10.1.2. Let $A$ be a p-complete commutative algebra object in $D(\mathrm{Fun}(\mathrm{CAlg}_{\mathrm{\mathbb{F}_p}}^{\mathrm{reg}},\mathrm{Ab}))$ equipped with an isomorphism $u_0\colon \Omega^*_{(-)}\xrightarrow{\sim} A\otimes^L\mathbb{F}_p$. Then $u_0$ lifts uniquely to an isomorphism $R\Gamma_{\mathrm{crys}}\xrightarrow{\sim} A$.
It seems to me that $A=\hat{\Omega}^*_{W(-)}$ already satisfies the assumption of the theorem. If so, what is the motivation for taking "strictification"? If not so, where am I misunderstanding?
