Q. about monoidal functors

Suppose _A_ is a symmetric monoidal category in the sense of the Eilenberg-Kelley La Jolla paper, and T: _A_ --> _A_ a monoidal functor. What, if anything, is known, where τ: X ⊗ Y --> Y ⊗ X is the symmetry structure on the (symmetric) tensor product ⊗, as to whether [T_X,Y: TX ⊗ TY --> T(X ⊗ Y)] and [T(τ_X,Y): T(X ⊗ Y) --> T(Y ⊗ X)] have the same composition as have [τ_TX,TY: TX ⊗ TY --> TY ⊗ TX] and [T_Y,X: TY ⊗ TX --> T(Y ⊗ X)] ? TIA for any relevant information and/or references thereto. Cheers, -- Fred [For admin and other information see: http://www.mta.ca/~cat-dist/ ]