A natural transformation Φ_Y :B(A Y)->Y is called unital if the leftmost diagram above commutes. Similarly, a natural transformation Ψ_Y :Y->A(B Y) is called unital if the diagram on the right-hand side above commutes. Note that in these definitions, A, B, and Y are all objects in a tensor category ℭ, I is the neutral (or identity) object in ℭ, and the juxtaposition A B is shorthand for the tensor product A×B in ℭ. What's more, the subscripts attached to the transformations Φ = {Φ_Y} and Ψ = {Ψ_Y} denote the components of the functors (indexed with respect to the objects in ℭ) in question.

category | category theory | functor | morphism | natural isomorphism | natural transformation | object | tensor category