I JUST REALIZED SOMETHING!!!

Claim: Associativity comes from functor composition.

**Proof:** Given any functor \\(F: \mathcal{C} \to \mathcal{D}\\), we can produce the following,

\\[

F(f(g(x)) \\\\

= F((f \circ g)(x))\\\\

= F(f \circ g) \circ F(x) \\\\

= F(f)\circ F(g \circ x) \\\\

= F(f\circ g \circ x)

\\]

Now as you can plainly see,

if you set \\(F\\) to \\(Id\_\mathcal{C}\\),

you'll get associativity in the category for free.

Q.E.D.

Claim: Associativity comes from functor composition.

**Proof:** Given any functor \\(F: \mathcal{C} \to \mathcal{D}\\), we can produce the following,

\\[

F(f(g(x)) \\\\

= F((f \circ g)(x))\\\\

= F(f \circ g) \circ F(x) \\\\

= F(f)\circ F(g \circ x) \\\\

= F(f\circ g \circ x)

\\]

Now as you can plainly see,

if you set \\(F\\) to \\(Id\_\mathcal{C}\\),

you'll get associativity in the category for free.

Q.E.D.