What are some criticisms of HEXACO from the Big Five camp?
We had a discussion of the pros and cons of the HEXACO model in a recent article that draws on arguments and evidence presented by the &quot;heavyweights&quot; of the Big 5 (Anglim, &amp; O'Connor, 2019).
Here is the quote of the paragraph that focusses specifically on the critiques:

Although the HEXACO model is receiving growing support, the necessity
of a sixth broad trait, as well as the nature of this trait are
subjects of current debate.

First, other prominent six factor
inventories including the Hogan Personality Inventory (Hogan &amp; Hogan,
1992), the Six Factor Personality Questionnaire (Jackson, Paunonen, &amp;
Tremblay, 2000), and Saucier's Big Six (Saucier, 2009) have not
converged on the same sixth factor in the same way that has broadly
occurred with the Big Five (De Raad, Barelds, Mlačić, et al., 2010).
Second, while several lexical studies have broadly supported the
HEXACO model, the support is far from universal.
Third, Saucier and
Ostendorf (1999) have argued that any broad trait in addition to the
Big Five will correlate highly with one of the existing Big Five, and
honesty-humility correlates relatively highly with agreeableness (r =
.67 in Gaughan, Miller, &amp; Lynam, 2012). Honesty-humility also
correlates reasonably well with straightforwardness and modesty facets
of agreeableness in the NEO-PI-R, suggesting that a comprehensive
facet-level model using the Big Five may provide similar predictive
benefits (Costa &amp; McCrae, 1995; Gaughan et al., 2012).
Fourth,
Barford, Zhao, and Smillie (2015) mapped the Big Five and HEXACO
traits to a common space provided by the interpersonal circumplex, and
found that honesty-humility occupied a fairly similar space to the
politeness aspect of B5 agreeableness.
Finally, Condon (2017) assessed
the covariance structure of approximately 700 statement-based
personality test items drawn from the most commonly used public-domain
personality measures (including an approximately equal representation
of items from prominent five and six-factor measures). While he found
support for a large number of narrow traits, factor analysis suggested
five broad traits, and when six factors were extracted, the sixth
factor did not correspond to honesty-humility.


References:

Anglim, J., &amp; O'Connor, P. (2019). Measurement and Research Using the Big Five, HEXACO, and Narrow Traits: A Primer for Researchers and Practitioners. Australian Journal of Psychology, 71, 16-25. https://psyarxiv.com/a78g2/download