In the article of A. Joyal and R. Street:

*Braided tensor categories*, (Adv. Math. N.102 (1993), no. 1, 20-78)

they define a **tensor object** (or **pseudo.monoids**) in a monoidal 2-category $\mathcal{C}$ (a monoidal structure on the underling category where all data are 2-natural too) as a 6-pla $(A, m , e, \alpha, \lambda, \rho)$ where
$A\in\mathcal{C}$, $m: A \otimes A \to A$, $e: I \to A$, and 2-isomorphisms:

$\alpha : m \ast (m \otimes 1) \Rightarrow m \ast (1 \otimes m) : A \otimes A \otimes A\to A $, $\rho: m\ast Ae \Rightarrow 1_A $, $\lambda: m\ast eA \Rightarrow 1_A $ where $Ae: A \cong A \otimes I \xrightarrow{1\otimes e} A\otimes A$ similarly for $eA$ and $AeA$ ecc. and we leave out canonical isomorphisms of $\mathcal{C}$.

With the following axioms:

**MA)** the composition:

$m\circ (m\otimes 1)\circ (m\otimes 1\otimes 1) \xrightarrow{m\circ( \alpha\otimes 1)} m\circ (m\otimes 1)\circ (1\otimes m\otimes 1) \xrightarrow{\alpha\otimes 1} m\circ (1\otimes m)\circ (1\otimes m\otimes 1) \xrightarrow{m\circ (1\otimes \alpha)}m\circ (1\otimes m)\circ (1\otimes 1\otimes m)$

is equal to:

$m\circ (m\otimes 1)\circ (m\otimes 1\otimes 1) \xrightarrow{\alpha\otimes 1} m\circ (1\otimes m)\circ (m\otimes 1\otimes 1) \xrightarrow{=} m\circ (m\otimes 1)\circ (1\otimes 1\otimes m) \xrightarrow{\alpha\otimes 1} m\circ (1\otimes m)\circ (1\otimes 1\otimes m)$

**MU)** the composition:

$m\circ(m\otimes 1)\circ (AeA) \xrightarrow{m\circ(\rho^{-1}\otimes 1)} m \xrightarrow{m\circ \lambda} m\circ(1\otimes m)\circ AeA$ is equal to: $m\circ(m\otimes 1)\circ (AeA) \xrightarrow{\alpha\circ AeA} m\circ(1\otimes m)\circ AeA$

If we consider a pseudo-monoid in the monoidal 2-category $Cat$ (with cartesian monoidal structure) we get a usual (small) monoidal category, and by G. Kelly theorems all diagrams of canonical isomorphisms are commutative, but this is like saying that all the (two dimensional) diagram built by canonical 2-cells are commutative (or coherent).

**I ask**: Is this true in general? Are all the cell diagrams of a pseudomponoids (built using only the canonical $\alpha, \rho, \lambda$ the tensor composition, and the canonical isomorphisms of $\mathcal{C}$) coherent?

I just know that a pseudo-monoid is a particular tricategory (with the theorem of triequivalence to a Gray-enriched category), but in this full-generality we have only a "weak coherence" (no all diagrams commute).

**EDIT**:
Edit: Given a diagram of canonical cells on $\mathcal{C}$ one could think to use some "kind of Yoneda lemma" for traslate the question on $Cat$, the fault is that $[-, X]: \mathcal{C}^{op}\to Cat$ don't preserve (neither has a some kind of correlation to) the monoidal product $\otimes$. But if $X$ is a $\otimes$-monoid then

$[-, X]: \mathcal{C}^{op}\to Cat$ is a monoidal functors

(from the morphisms-dual (the some monoidal product) of $\mathscr{C}$ to the monoidal cartesian structure of $Cat$)

and if applied on a pseudo-monoid of $\mathcal{C}$ maps it in a pseudo-monoid of $Cat$ (diagrams (MF1), (MF2), (MF3) p. 473 of Kelly, Eilenberg "Closed CAtegories", then if applied to a cells diagrams (relative to a pseudo-monoid) we get a similar diagram (relative to pseudo-monoid image)

(the some if $X$ is a comonoid and consider $[X,-]: \mathcal{C} \to Cat$, if $X$ is a pseudo-monoid $[-, X]$ is a pseudo.monoidal functors i.e. the diagrams (MF1), (MF2), (MF3) commuting but 2-isomorphisms).

Now the functors $[X, -]$ ($X\in \mathcal{C}$ simply) are useful because they are collectively faithful (locally too) (then if their images is ever a coherent diagram the initial diagram is coherent), but in our situation we have to use only monoids $X$ (no simple objects).

**Then the second question** is: Has a monoidal 2-category $\mathcal{C}$ enough-monoids (i.e. the family of $[-, X]$ for $X$ monoid, is (locally) faithful) ?

If we request $\mathcal{C}$ representable as 2-category (i.e. with comma objects) has it enough-monoids?

**EDIT**:

I did this partial answer (enough for many results):

Given a 2-category $\mathscr{A}$ and a object $G \in C$ we define the category $\mathscr{A} \searrow G$ with object of type $(A, a)$, $a: A \to G$ and morphisms of type $(f, \phi ): (A, a) \to (B, b)$, $f: A \to B$, $\phi : a \Rightarrow b \circ f$ and natural composition and unities. Let $(G, \alpha, \lambda , \rho, m, e)$ a pseudomonoid of a monoidal 2-category $\mathscr{C}$, and consider the full subcategory $ \widetilde{G} \subset \mathscr{C} \searrow G$ where objets are defined by induction as:

1) $(I, e)$ and $(G, 1_G)$ are object of $\widetilde{G}$.

2) if $(X, x), (Y, y)$ are object of $\widetilde{G}$ then so is also their monoidal product $(X, x) \otimes (Y, y)$ defined as $(X \otimes Y, m\circ (x \otimes y))$.

Now $\widetilde{G}$ has a monoidal structure, with $(I, e)$ as unity object, monoidal product defined as above, with canonical isomorphism defined by induction using the elements $\alpha, \lambda , \rho, m, e$ (I seems that work, it is tedious but no hard).

Then we have a partial results: *all diagram of cells (having a final vertex $G$ corresponding to canonical diagrams of $\widetilde{G}$) are coherent.*

If we suppose that $\mathscr{C}$ has a symmetry, can generalize a bit because a monoidal product of copies of $G$ or $I$ become the monoidal product of the pseudo.monoids $G$ or $I$, and a reasoning similar to above (make a new induction building, that start from a object already built with the prior induction) concludes the proof.