A&A's Concession to Complementarity

Here is what Sutton, et al., have to say about the A&A concession to complementarity:

After quoting Clark (1998) and Sutton’s (2010) exposition of the complementarity principle, Adams and Aizawa write ‘We agree with this completely’. We are delighted that they accept the basic claims of the complementarity argument.

But, that doesn't get our view right.  We concede that there is a complementarity between brain processes and environmental processes, but we do not concede an argument from complementarity of brain and environmental processes to extended cognitive processes.  So, we do not accept the basic claims of the complementarity argument.  Consider the text of The Bounds of Cognition,

“Second-wave” extended cognition, however, is based on what Sutton calls a ‘complementarity principle”: 

in extended cognitive systems, external states and processes need not mimic or replicate the formats, dynamics, or functions of inner states and processes.  Rather, different components of the overall (enduring or temporary) system can play quite different roles and have different properties while coupling in collective and complementary contributions to flexible thinking and acting” (Sutton, forthcoming)

We agree with this completely.  Only, we do not think this in any way supports the hypothesis that cognitive processes extend from the brain into the body and environment.  Many of the ideas we have developed in previous chapters should make it clear why.  (Adams & Aizawa, 2008, p. 145).

The A&A view is that there is complementarity, but that's no reason to believe in HEC.  One should, however, bear in mind that A&A draw a distinction between the hypothesis of extended cognitive systems and the hypothesis of extended cognition (or hypothesis of extended cognitive processes).