3.2.7 The Monoidal Category of Sets and Coproducts

The category Sets admits a closed symmetric monoidal category structure consisting of:

The Pentagon Identity
Let W, X, Y and Z be sets. We have to show that the diagram
commutes. Indeed, this diagram acts on elements as
and therefore the pentagon identity is satisfied.

The Triangle Identity
Let X and Y be sets. We have to show that the diagram

commutes. Indeed, this diagram acts on elements as

and therefore the triangle identity is satisfied.

The Left Hexagon Identity
Let X, Y, and Z be sets. We have to show that the diagram

commutes. Indeed, this diagram acts on elements as

and thus the left hexagon identity is satisfied.

The Right Hexagon Identity
Let X, Y, and Z be sets. We have to show that the diagram

commutes. Indeed, this diagram acts on elements as

and thus the right hexagon identity is satisfied.


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