In 2-category theory, there is a notion of "lax idempotent 2-monad" $M$ on a 2-category, for which the multiplication $m: M M \to M$ is left adjoint to the unit $u M: M \to M M$. A typical sort of example is a cocompletion monad with respect to some class of colimits. For example, consider the 2-category $Pos$ of posets; the 2-category $Sup$ of sup-lattices is 2-monadic over $Pos$, where the free sup-lattice generated by a poset $P$ is the sup-lattice of poset maps $P^{op} \to \mathbf{2} = \{0 \leq 1\}$ (equivalently, the sup-lattice of downward closed subsets). In this case the unit of the 2-monad is a Yoneda embedding $y_P: P \to [P^{op}, \mathbf{2}]$, and the 2-monad multiplication is

$$[y_P^{op}, \mathbf{2}]: [[P^{op}, \mathbf{2}]^{op}, \mathbf{2}] \to [P^{op}, \mathbf{2}].$$

What turns out to be true always is that $M u: M \to M M$ is in turn left adjoint to $m: M M \to M$.

Just as the augmented simplicial category $\Delta$ of finite ordinals (including the empty ordinal), as a monoidal 1-category equipped with a monoid = 1-element ordinal, is initial among monoidal categories equipped with a monoid, we can make an analogous statement for $\Delta$ as a 2-category (hom-sets being made into poset categories since ordinal maps are ordered among themselves). Namely, we can construct syntactically the initial monoidal 2-category equipped with a lax idempotent monoid, which will have invertible triangulator isomorphisms as part of the data, with suitable coherence conditions. And what turns out to be true is that this initial structure is equivalent (as a monoidal 2-category) to $\Delta$. Notice that 2-cell isomorphisms of $\Delta$ are automatically identities, meaning that the built-in coherence conditions on the initial structure can be "strictified".

The nLab has some information and references for lax idempotent monads which may be useful to you. I'll try to track down a reference for this 2-categorical fact about $\Delta$, but you can check by hand the presence of a lax idempotent (aka "Kock-Zöberlein") monoid structure on the 1-element ordinal.

**Edit:** Perhaps a canonical reference for this material is Street's Fibrations in Bicategories, particularly section 2.

Higher topos theory. – Zhen Lin Apr 24 at 21:13