Let $k$ be a commutative base ring. We have a category $\operatorname{Mod}_k$ of $k$-modules and a category $\operatorname{grMod}_k$ of $\mathbb{Z}$-graded $k$-modules. Both of these have monoidal structures, and the first has a canonical symmetric monoidal structure. The second category has (at least) two symmetric monoidal structures/braidings $\beta : V \otimes W \to W \otimes V$: the "non-oriented" braiding given on homogeneous simple tensors by $\beta(v \otimes w) = w \otimes v$ and the "oriented"/"Koszul" braiding given on homogeneous simple tensors by $\beta(v \otimes w) = (-1)^{|v| \cdot |w|}w \otimes v$. I'm writing some notes where I try to get the signs right in DG/graded algebra and to maintain consistency I'm trying to do everything using the Koszul braiding.
Now consider the tensor algebra $T(V)$ for a $k$-module $V$. This can be defined without ambiguity as the free graded algebra on the graded module $\Sigma V$ ($V$ concentrated in degree $1$). We can use the universal property of the free algebra to define counit and comultiplication operations, the counit being the map $T(V) \to k$ induced by the zero map $\Sigma V \to k$ and the comultiplication being induced by the map $\Sigma V \to T(V) \otimes T(V)$ sending $v$ to $v \otimes 1 + 1 \otimes v$. Of course to define this we need an graded-algebra structure on the tensor product $T(V) \otimes T(V)$. In any symmetric monoidal category the tensor product of the underlying objects of a pair of monoids inherits a monoid structure, but this crucially requires the braiding. More concretely if $A, B$ are graded algebras we can either define the multiplication (on homogeneous simple tensors) as $(a_1 \otimes b_1) \cdot (a_2 \otimes b_2) = a_1 a_2 \cdot b_1 b_2$ or as $(a_1 \otimes b_1) \cdot (a_2 \otimes b_2) = (-1)^{|b_1| |a_2|} a_1 a_2 \cdot b_1 b_2$. These result in different comultiplications on the tensor algebra. The wikipedia article for the tensor algebra gives a closed form for the (unoriented) comultiplication as $$\Delta(v_1 \cdots v_m) = \sum_{p+q=m} \sum_{\sigma \in \operatorname{Shuffle}(p, q)} = (v_{\sigma(1)} \cdots v_{\sigma(p)}) \otimes (v_{\sigma(p+1)} \cdots v_{\sigma(m)}).$$ Purely by guesswork I would expect the expression for the Koszul-comultiplication to be the same as above but with signs of the $\sigma$ inserted (like for a determinant).
Is this Koszul-comultiplication in the literature somewhere? Do any of the desiderata of the unoriented-comultiplication fail for this new coalgebra structure? E.g. if I wanted to define the symmetric or exterior hopf algebras as the quotient by the usual ideals, are those ideals are biideals in this new monoidal structure?
