SUMMARY: Observing that monoids in a monoidal category are identified with lax.functors (with domain 1), I tried to generalize this argument wanting to get a skew-Monoidal-category as (tri)lax.functor (with domain 1) in the category 2- monoidal (Cartesian) $CAt$ considered as tricategory. But there is a problem with the axiom.
Let $1$ the final 2-category (one arrow and one cell, both identity), and let $\mathcal{C}$ a monoidal category considered also as a bicategory with just one object. Then monoids $(A, m, e)$ (where $m: A\otimes A\to A,\ e: I\otimes A\to A$ as usual ) of $\mathcal{C}$ are identified by lax-functors $A: 1 \to \mathcal{C}$ (easy check). And a monoid in the monoidal cartesian category $CAt$ is a strict monoidal category
Now a "laxification" of monoids is a lax-monoids, we can do it form the notion of tensor object given by R.Street and A. Joyal in [BTC], def.5.1 p. 61-2, taking $\rho$ in reverse direction and change the second unitary axiom, and add other unitary axioms that we label as $(2), (3), (4), (5)$ in manner that a lax.monoid in monoidal the 2-monoidal cartesian category $CAt$ is a skew-monoidal category as defined in $[RUS]$ with the natural correspondence by axioms the aximos $(2), (3), (4), (5)$ in that article. And these axioms are indipendent (as proved in $[RUS]$).
I ask if a (tri)lax.funtor $H: 1 \to \mathscr{C}$ (where $\mathscr{C}$ is a tricategory) is a "lax.monoid", assuming that in the axiom $(HTD3)$ of [CT] p. 15 we have a strict 2-functor. But if $\mathscr{C}$ is the 2-monidal $CAt$ (consider as a tricateegory with just one object) we have only the axiom $(1)$ abd $(3)$ (see $RUS$ p.2-3) of skew-monoidal category.
it is not appropriate to add additional axiom's (similar to $(2), (4), (5)$ of [RUS]) in the definition of trifunctor of [CT]?
[RUS]: Remarks On Units Of Skew Monoidal Categories, Jim Andrianopoulos (http://arxiv.org/abs/1505.02048)
[BTC]: Braided tensor categories, André Joyal and Ross Street, Adv. Math. 102 (1993), 20-78.
[CT]: Coherence for Tricategories, Gordon, Power, Street, MAMS 117 (1995) no 558.
