  We study new coalgebra structures on the tensor product of two coalgebras $C$
and $D$ by twisting the tensor product coalgebra via a twist map $\Psi: C
\otimes D \rightarrow D \otimes C$. We deal with the general case in which the
counit of the tensor product coalgebra is deformed as well. Some classes of
such deformations are analyzed and a notion of equivalence of twists is
discussed. We also present the dual deformation of tensor product algebras and
provide examples.
