  We define self-distributive structures in the categories of coalgebras and
cocommutative coalgebras. We obtain examples from vector spaces whose bases are
the elements of finite quandles, the direct sum of a Lie algebra with its
ground field, and Hopf algebras. The self-distributive operations of these
structures provide solutions of the Yang--Baxter equation, and, conversely,
solutions of the Yang--Baxter equation can be used to construct
self-distributive operations in certain categories.
  Moreover, we present a cohomology theory that encompasses both Lie algebra
and quandle cohomologies, is analogous to Hochschild cohomology, and can be
used to study deformations of these self-distributive structures. All of the
work here is informed via diagrammatic computations.
