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- Let be the set and be the underlying field
- Let denote the multiplication and the unit of (for its algebra structure)
- Let denote the comultiplication and the counit of .
- Let be the unique linear map from to itself that sends each pure tensor to .
The compatibility conditions are as follows:
- Compatibility between multiplication and comultiplication:
- Compatibility between multiplication and counit:
under the canonical identification of with .
- Compatibility between comultiplication and unit:
- Compatibility between unit and counit: