My question is about monoidal categories. To motivate it, let me first recall something about group objects.
Assume you define a group object in a category $C$ with products by an object $G$ together with morphisms $G \times G \to G, G \to G, * \to G$, so that the diagrams commute which correspond to the group axioms. Then you want to generalize some elementary properties from usual groups ($C=Set$) to these group objects, but it is quite hard to write down the relevant diagrams. For example when you want to prove that there is a actually at most one unit $* \to G$, a morphism $G \to G'$ between group objects which respects the multiplication already respects the inversion and the unit, left-inverse implies right-inverse, etc. But all this may be reduced to the case $C=Set$ by using the Yoneda-Lemma and the the definition of a group object as an object together with a factoriation of its hom-functor over the category of usual groups. Then these calculations are easy and you don't have to produce all these diagrams.
My question is: Is there a similar definition for a monoidal category? Specifically, I want to see a neat (diagram-free?) proof of Lemma 3.2.5 in this note (Pareigis' lectures on quantum groups), which is intuitive and does not come up with diagrams without motivating them. (For me, "We tried to prove it and finally this diagram worked" is no intuition.) The lemma implies for example that the endomorphism monoid of the unit object of a monoidal category is abelian, which is quite surprising (for me). This Lemma is just one example. It seems to me that monoidal categories are "weak monoids in the 2-category Cat", but I don't see yet if this description actually simplifies these diagrams.
Best Answer
The kind of thing you are looking for applies not just to monoidal categories, but to bicategories, and it is called the bicategorical Yoneda lemma. If $B$ is a small bicategory, one may form the strict 2-category $[B^{op}, Cat]$ consisting of weak 2-functors (aka homomorphisms), pseudonatural transformations, and modifications from $B^{op}$ to $Cat$. Then there is a Yoneda embedding
$$y: B \to [B^{op}, Cat]$$
sending an object $b$ to $\hom(-, b)$, and the mapping of $B$ onto its image is a bi-equivalence. The image however is a strict 2-category, and hence we get from this a coherence theorem which assures us that all definable diagrams (in say the free monoidal category generated by a discrete category) commute. This line of thinking was introduced by Street in his paper Fibrations in Bicategories (for which there is also a correction).
This is really the modern point of view on coherence in monoidal categories. An exposition which focuses on monoidal categories can be found (I believe) in Braided Tensor Categories by Joyal and Street.
It is true that monoidal categories can be described as weak monoids, but this by itself doesn't solve coherence issues.