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Deformation of tensor product (co)algebras via non-(co)normal twists

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 $Ψ: C \otimes D \rightarrow D \otimes C$. 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.