Let $A = \bigoplus^n_{i=0} A_i,B = \bigoplus^b_{i=0} B_i $ be a graded modules over the same commutative ring $R$ . The twisting isomotphism $\tau_{A,B} : A \otimes B \to B \otimes A$ is defined on elementary tensors formed by homgeneous elements $a \in A_i$ and $b \in B_j$ by $\tau_{A,B}(a \otimes b) = (-1)^{ij}(b \otimes a)$, and then extended by linearity on the whole tensor product.
A graded coalgebra $A$ is said to be skew-cocommutative if the twisting isomorphism preserves coproducts, that is $\tau_{A,A} \circ \Delta = \Delta$. If $(A_i,\Delta_i,\eta_i)^n_{i=1}$ is a finite collection of coalgebras, then their tensor product $\bigotimes^n_{i=1} A_i$ can be equipped skew tensor product coalgebra structure $\Delta$ defined by the relation on elementary tensors of homogeneous elements $a_i$ as $$ \Delta\left( \bigotimes^n_{i=1} a_i\right) = \sum_a (-1)^{I,J} \bigotimes^n_{i=1} a_{i,1} \otimes \bigotimes^n_{i=1} a_{i,2}. $$ Here I use Sweedler's notation. And for each summand $I = (\deg a_{i,1})^n_{i=1}$ and $J = (\deg a_{j,2})^n_{j=1}$ are multi-indices used to compute $(-1)^{I,J}$. This $(-1)^{I,J} = (-1)^N$, with $$N = \sum^n_{j=1} \sum^n_{i=j+1} J_j I_i.$$
And the counit is defined by
$$ \eta\left( \bigotimes^n_{i=1} a_i \right) = \prod^n_{i=1} \eta_i(a_i) $$
It can be proved that skew-tensor product of skew-cocommutative coalgebras is again a skew-cocommutative coalgebra. Moreover, I can prove that tensor product of cocommutative coalgebras is a product of these coalgebras in the category of ccocommutative coalgebras. The symmetry demands skew-tensor product to be the product in the category of skew-commutative coalgebras.
Is my conjecture true? Is the skew-tensor product is the finite product in the he category of skew-commutative coalgebras?