If $V$ is a vector space, let us consider the tensor coalgebra $TV=\oplus_{k=0}^\infty V^{\otimes^k}$ with coproduct given by
$$\Delta (x_1\otimes \dots \otimes x_n):= \sum_{i=0}^{n}(x_1\otimes\dots\otimes x_{i-1})\bigotimes (x_i\otimes\dots\otimes x_{n}).$$
Its reduced version is $\overline{TV}=\oplus_{k=1}^\infty V^{\otimes^k}$ with coproduct $\bar{\Delta}$ given by replacing $1$ by $0$ 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.
