This might be inappropriate for the MO-level. If so I'll delete it...

Suppose $V$ is a $\mathbb{Z}$-graded vector space and $\overline{T}(V):=V \oplus V\otimes V \oplus \otimes^3 V \ldots$ is the 'reduced tensor power' of $V$ in the graded sense.i.e:

$V\otimes V:=\oplus_{z\in\mathbb{Z}}\oplus_{p+q=z}V_p\otimes V_q$

We don't see $T(V)$ as a (co)algebra, just as a plain graded vector space.

Then on a first "empirical observation" I would say, that if we apply the reduced tensor functor again, i.e make $\overline{T}(\overline{T}(V))$ then this is naturally isomorphic to $\overline{T}(V)$ because "all tensor powers are already there".

So this should be a consequence of the 'freeness' of $\overline{T}(V)$, but I can't see how this can be proofen.