TITLE: Gurski's Definition of a strict functor of tricategories
QUESTION [8 upvotes]: In Gurskis definition (page 32 of his thesis) of a strict functor $F$ of tricategories he requires that

$F$ maps the adjoint equivalences $a,l,r$ in the source tricategory to the same adjoint equivalences in the target tricategory.

This means for example $F (a_{fgh}) = a_{FfFgFh}$. (Note that these morphisms are indeed parallel, because of the strictness of $F$.)
However it is not required, that the modifications in the definition of a tricateory $(\pi,\lambda,\mu,\rho)$ are preserved as well. For example 
$ F (\pi _{fghk}) $ is not required to be equal to $ \pi_{FfFgFhFk} $.
So my question is:
Why isn't it required that $(\pi,\lambda,\mu,\rho)$ are preserved?
Is this because this is already implied by the other requirements?
Or are there strict functors which doesn't preserve the modifications $(\pi,\lambda,\mu,\rho)?$
EDIT
Let me draw some parallel to the definition of strict functors between bicategories: ($*$ denotes composition along objects)
There it is only required that the constraint-2-cells
$$ F(f)*F(g) \to F(f*g) $$
and
$$ 1_{Fa} \to F(1_a) $$ are identities.
That $a,l,r$ are preserved is not demanded by definition, but it follows easily from the functor axioms.
Therefore one might hope that in the case of a strict functor of tricategories it follows somewhat analoge that the "highest" constraint data $\pi,\lambda,\mu,\rho$ are preserved, although it is not demanded by definition.

REPLY [2 votes]: The axioms in the definition of a non-strict trihomomorphism (Definition 3.3.1, page 31-32) already require that the modifications $\pi,\lambda,\mu,\rho$ are preserved "up to" the coherences $\pmb{\chi}, \pmb{\iota}, \omega,\gamma,\delta$.  The definition of a strict functor (3.3.3, p33) then requires that $\pmb{\chi}$ and $\pmb{\iota}$ are identities while $\omega,\gamma,\delta$ are "as close to identities as possible" (they can't literally be identities because their domain is not equal to their codomain), which therefore implies that $\pi,\lambda,\mu,\rho$ are "preserved as much as possible".