Is there anything more than a superficial similarity between the following?
The Alexander-Whitney coproduct $\Delta$ on the tensor algebra $\bigotimes^\bullet V$ of a vector space $V$ is defined by the equalities $$ \Delta(1) = 1,\quad \Delta(v) = v\otimes 1 + 1\otimes v $$ and compatibility with the product $\nabla = {-}\otimes{-}\,$.
A derivation $D$ on a (unital associative) algebra $(A,\nabla,\eta)$ is defined as a linear map $A\to A$ satisfying the Leibniz rule $$ D\circ\nabla =\nabla\circ (D\otimes\mathrm{id} +\mathrm{id}\otimes D); $$ dually, a coderivation $D$ on a coalgebra $(C,\Delta,\epsilon)$ is a linear map $C\to C$ satisfying the co-Leibniz rule $$ \Delta\circ D = (D\otimes\mathrm{id} +\mathrm{id}\otimes D)\circ\Delta\,. $$
A subspace $I\subseteq C$ of a coalgebra $(C,\Delta,\epsilon)$ is a coideal if $$ \epsilon(I) = 0,\quad \Delta(I)\subseteq I\otimes C + C\otimes I. $$