If $V$ is a vector space, let us consider the tensor coalgebra $TV=\bigoplus\limits_{k=0}^\infty V^{\otimes^k}$ with coproduct given by $$\Delta (x_1\otimes \dots \otimes x_n):= \sum\limits_{i=0}^{n}(x_1\otimes\dots\otimes x_i)\bigotimes (x_{i+1}\otimes\dots\otimes x_{n}) .$$

Its reduced version is $\overline{TV}=\bigoplus\limits_{k=1}^\infty V^{\otimes^k}$ with coproduct $\bar{\Delta}$ given by replacing $\sum\limits_{i=0}^{n}$ by $\sum\limits_{i=1}^{n-1}$ in the previous formula. In other words: $$\Delta (x)=\bar{\Delta}(x)+1\otimes x + x\otimes 1 .$$

It is well known that a coderivation of $\overline{TV}$, i.e. $Q\in L(\overline{TV},\overline{TV})$ satisfying $(Q\otimes Id + Id\otimes Q)\circ \Delta=\Delta \circ Q$, is determined by its corestrictions, i.e. by $\Pi_V\circ Q$, where $\Pi_V\colon \overline{TV}\rightarrow V$ is the canonical projection on $V$.

What I am looking for is a reference for a proof of the same statement for $TV$, the non reduced tensor coalgebra.