Let $K$ be a complete discrete valued field of characteristic zero with perfect residue field $k$ of characteristic $p>0$, $\mathcal{O}_K$ its ring of integers, $C$ the completion of an algebraic closure of $K$ and $\mathcal{O}_C$ its ring of integers. Fontaine defines in "Le corps des périodes p-adiques" (see loc.cit, remark 1.2.4.(c))
$$\rm{A}_{inf}(\mathcal{O}_C|\mathcal{O}_K)$$
as the completion of $\mathcal{O}_K\otimes_{W(k)}W(\mathcal{O}_C^{\flat})$ for the topology defined by the ideal ($p$)+ker($\theta$), for a certain map
$$\theta :\mathcal{O}_K\otimes_{W(k)}W(\mathcal{O}_C^{\flat}) \to \mathcal{O}_C$$
where $W$ denotes Witt vectors, and $\mathcal{O}_C^{\flat}$ is the tilt of $\mathcal{O}_C$, that is, the projective limit of $...\to \mathcal{O}_C/p\mathcal{O}_C\to \mathcal{O}_C/p\mathcal{O}_C$ where the maps are Frobenius.
Later, in 1.3.3, it is claimed that
$$\rm{A}_{inf}(\mathcal{O}_C|\mathcal{O}_K)=\mathcal{O}_K\otimes_{W(k)}W(\mathcal{O}_C^{\flat})$$
(that is, there is no need of completing).
Why is this last claim true?