How do the coherence conditions for a monoidal category imply "associativity of the monoidal product" Intuitively, I would say that "associativity of the monoidal product" should mean:


*

*for all objects $A,B,C$, there is a natural isomorphism so that $(A\ast B)\ast C \cong A\ast (B\ast C)$, and

*for all morphisms $f,g,h$, it holds that $(f\ast g)\ast h=f\ast (g\ast h)$. 
Obviously, nr 2 doesn't make sense, given that the objects of $(f\ast g)\ast h$ and $f\ast (g\ast h)$ merely have to be isomorphic but not identical.
Is it true that the coherence conditions imply (or even are intended to imply) something like a weaker version of nr 2? If not, then how do the coherence conditions capture the intuitive notion that the monoidal product is "associative"?
 A: The closest thing to your second condition that we could ask in a monoidal category is that the the two terms of your equation become equal if you compose them with isomorphisms
$$\alpha_{A,B,C}:(A\ast B)\ast C \to A\ast (B\ast C)$$
and
$$\alpha_{A',B',C'}:(A'\ast B')\ast C' \to A'\ast (B'\ast C')$$
to compensate the fact that their domains and codomains don't match : this would give you the condition
$$(f\ast( g\ast h))\circ \alpha_{A,B,C}=\alpha_{A',B',C'}\circ ((f\ast g)\ast h).$$
Asking this to hold for all $f,g,h$ (and the isomorphisms $\alpha_{A,B,C}$ to depend only on the objects) is equivalent to asking that $(\alpha_{A,B,C})_{A,B,C\in Ob(\mathcal{C})}$ is a natural isomorphism, from the functor $(\_\ast \_)\ast \_$ to the functor $\_\ast (\_\ast \_)$, and this is part of the definition of a monoidal category; but that's not part of the coherence conditions.
The coherence conditions are intended to make sure that there is no ambiguity when you're looking for an isomorphism $(A\ast (B\ast (C\ast D)))\to (((A\ast B)\ast C)\ast D)$, for example : a priori you could move the parenthesis in different ways, and you want to make sure that the isomorphism you get at the end is always the same.
