Part : Cat → Cat
Part C = C^{● → ○}

● represents the total space
○ represents the set of indices

For our purposes C := Set.

⊔ : Part C → C
⊔ = composition with {●} → {● → ○}

D ⊣ ⊔ ⊣ ⊤

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The refinement of two partitions  ● → ○₁ and ● → ○₂
with the same total space is ● → ○₁ × ○₂.