Lemma 7.8.3. Let $\mathcal{C}$ be a category. Let $\mathcal{U} = \{ \varphi _ i : U_ i \to U\} _{i\in I}$, and $\mathcal{V} = \{ \psi _ j : V_ j \to U\} _{j\in J}$ be two families of morphisms with the same fixed target.

1. If $\mathcal{U}$ and $\mathcal{V}$ are combinatorially equivalent then they are tautologically equivalent.

2. If $\mathcal{U}$ and $\mathcal{V}$ are tautologically equivalent then $\mathcal{U}$ is a refinement of $\mathcal{V}$ and $\mathcal{V}$ is a refinement of $\mathcal{U}$.

3. The relation “being combinatorially equivalent” is an equivalence relation on all families of morphisms with fixed target.

4. The relation “being tautologically equivalent” is an equivalence relation on all families of morphisms with fixed target.

5. The relation “$\mathcal{U}$ refines $\mathcal{V}$ and $\mathcal{V}$ refines $\mathcal{U}$” is an equivalence relation on all families of morphisms with fixed target.

Proof. Omitted. $\square$

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