Mac Lane coherence theorem

In category theory, a branch of mathematics, Mac Lane coherence theorem states, in the words of Saunders Mac Lane, “every diagram commutes”. More precisely (cf. ), it states every formal diagram commutes, where "formal diagram" is an analog of well-formed formulae and terms in proof theory.

Counter-example
It is not reasonable to expect we can show literally every diagram commutes, due to the following example of Isbell.

Let $$\mathsf{Set}_0 \subset \mathsf{Set}$$ be a skeleton of the category of sets and D a unique countable set in it; note $$D \times D = D$$ by uniqueness. Let $$p : D = D \times D \to D$$ be the projection onto the first factor. For any functions $$f, g: D \to D$$, we have $$f \circ p = p \circ (f \times g)$$. Now, suppose the natural isomorphisms $$\alpha: X \times (Y \times Z) \simeq (X \times Y) \times Z$$ are the identity; in particular, that is the case for $$X = Y = Z = D$$. Then for any $$f, g, h: D \to D$$, since $$\alpha$$ is the identity and is natural,
 * $$f \circ p = p \circ (f \times (g \times h)) = p \circ \alpha \circ (f \times (g \times h)) = p \circ ((f \times g) \times h) \circ \alpha = (f \times g) \circ p$$.

Since $$p$$ is an epimorphism, this implies $$f = f \times g$$. Similarly, using the projection onto the second factor, we get $$g = f \times g$$ and so $$f = g$$, which is absurd.