Q. about monoidal functors

["Fred E.J. Linton" <fejlinton@usa.net>]

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/ ]