# Monoidal Category (Coherence Conditions)

In the definition of a monoidal category below, can someone please explain the idea behind the coherence conditions, especially the pentagon diagram's construction. Why do we need four elements A,B,C, and D in category C? Is the idea to try and cover all possible associative groupings for all 4 objects in C?

For ordinary monoids, we know that general associativity follows from the basic $(ab)c = a(bc)$ form. (and similarly for unit introduction/elimination)
This is basically enough to prove that you can construct an arbitrary coherence morphism using just associators $\alpha_{ABC} : (A \otimes B) \otimes C \to A \otimes (B \otimes C)$ along with the two unitors, the monoidal product, and composition. This allows the monoidal structure to be defined using just these three natural isomorphisms.