Let's suppose we have two noncommutative graded k-coalgebras $C_1$ and $C_2$ with respective admissible filtrations (i.e $F_{0}C_i=0$ and $\mathrm{colim}_k F_kC_i=C_i$), I would like to know if there is an isomorphism $$\mathrm{Gr}(C_1\prod C_2)\cong \mathrm{Gr}(C_1)\prod\mathrm{Gr}(C_2)$$ where the filtration in $C_1\prod C_2$ is $F_k(C_1\prod C_2)=\displaystyle\sum_{p+q=k}{F_pC_1\prod F_q C_2}$. I saw the dual result for k-algebras in page 190 of the book Cogroups and Co-rings in Categories of Associative Rings, but without proof. Any suggestion, please?.
By the way, $C_1\prod C_2=C_1\oplus C_2\oplus (C_1\otimes C_2)\oplus (C_2\otimes C_1)\oplus (C_1\otimes C_2\otimes C_1)\oplus\cdots$
the summands are alternating tensor products.