[[Category of categories]]
# Simpson's lemma

Let $\cat C_{1} \cong \cat C_{2}$ be [[Isomorphism of categories|isomorphic categories]] such that

1. $\cat C_{1}$ is a [[category of categories]];
2. $\cat C_{2}$ is a [[category of categories]]; and
3. there exist categories $\cat M, \cat N \in  \cat C_{1}$ isomorphic to [[Interval category]] and [[Ordinal category|$\underline 3$]] respectively.[^walking]

Then every category $\cat A \in \cat C_{1}$ is isomorphic to a category $\cat B \in \cat C_{2}$ and vice versa.

  [^walking]: [[The walking]] [[morphism]] and composition respectively.

> [!check]- Proof
> Since functors $\cat M \to \cat A$ are precisely morphisms and functors $\cat N \to \cat A$ determine composition,
> it follows that the isomorphism class of $\cat A$ (as a category)
> is determined by the isomorphism class of $\cat C_{1}$.
> The same goes in the opposite direction. <span class="QED"/>

A corollary is that any [[Pseudoautistic category|pseudoautistic]] [[category of categories]] containing categories isomorphic to $\cat 2$ and $\cat 3$ is [[Autistic category|autistic]].[^1999]

  [^1999]: 1999\. [[Sources/@simpsonFOMRussellParadox1999|FOM: Russell paradox for naive category theory]]

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