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.