I have a question regarding an exercise(p.38, ex.5.9) from Aluffi's Algebra textbook.

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> Let ***C*** be a category with products. Find a reasonable candidate for the universal property that the product $A \times B \times C$ ought to satisfy, and prove that both $(A \times B) \times C$ and $A \times (B \times C)$ satisfy the universal property.Deduce that $(A \times B) \times C$ and $A \times (B \times C)$ are necessarily isomorphic.
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$A \times B \times C$ is defined an object of category ***C*** satisfying( together with morphisms $\pi''\_A:A \times B \times C \to A, \pi''\_B:A \times B \times C \to B$ and $\pi''\_C:A \times B \times C \to C$ the following universal property:
$\forall$ objects $X$ of the category ***C*** and morphisms from $X$ to $A(f\_0),B(g\_0)$ and $C(h\_0)\ \exists!$ morphism $\omega$ such that 

$\pi''\_A\omega=f\_0\\
\pi''\_B\omega=g\_0 \ \ (1)\\
\pi''\_C\omega=h\_0$

Now, I want to prove that $(A \times B) \times C$ also satisfies this property together with morphisms $\pi\_A\pi'\_{A \times B}, \pi\_B\pi'\_{A \times B}$ and $\pi'\_C$.

$(A \times B) \times C$ has it's own property. $\forall$ objects $X\_1$(of the category ***C***) and morphisms $f\_1:X\_1 \to A \times B, h\_1:X\_1 \to C \ \exists! \ \phi:X\_1 \to (A \times B) \times C$ such that the following diagram
$$ \begin{array}{ccccc}
A \times B & \stackrel {\pi'\_{A \times B}} \leftarrow & (A \times B) \times C & \stackrel{\pi'\_C} \to & C \\
 & {}\_{f\_1} \nwarrow & \ \ \ \ \ \uparrow {}\_{\phi} & \nearrow {}\_{h\_1} & \\
 & & X\_1 & & 
\end{array} $$ 
commutes.

And, finally, we have the universal property of $A \times B:
\forall X\_2$ and morphisms $f\_2:X\_2 \to A, g\_2:X\_2 \to B$ we have a unique morphism $\upsilon:X\_2 \to A \times B$ such that the following diagram
$$ \begin{array}{ccccc}
A & \stackrel {\pi\_A} \leftarrow & A \times B & \stackrel{\pi\_B} \to & B \\
 & {}\_{f\_2} \nwarrow & \ \ \ \ \ \uparrow {}\_{\upsilon} & \nearrow {}\_{g\_2} & \\
 & & X\_1 & & 
\end{array} $$ 
commutes.
From the universal properties of $A \times (B \times C)$ and $A \times B$ follows that $\forall$ objects $X$ and morphisms $f,g$ and $h$ we have a unique morphism $\psi:X \to (A \times B) \times C$ such that
$\pi\_A\pi'\_{A \times B}\psi = f\\\pi'\_C\psi = h$.

Now, how do we prove that an analogue of $(1)$ goes for $(A \times B) \times C$ as well?