Additive endomorphism satisfying a comultiplication condition

Definition
Let $$p$$ be a formal expression in $$\mathbb{Z}[t] \otimes_{\mathbb{Z}} \mathbb{Z}[t]$$; it can be written explicitly as a sum:

$$\sum_{(i,j) \in \mathbb{N}_0 \times \mathbb{N}_0} a_{ij}t^i \otimes t^j$$

where only finitely many of the $$a_{ij}$$s are nonzero.

Suppose $$R$$ is a non-associative ring with multiplication $$*$$ and $$f$$ is an endomorphism of the additive group of $$R$$. We say that $$f$$ satisfies the comultiplication condition for $$p$$ if the following holds:

$$f(x * y) = \sum_{(i,j) \in \mathbb{N}_0 \times \mathbb{N}_0} a_{ij}f^i(x) * f^j(y)$$