Warning 2.4.5.1.3 (01JN): Free Cocompletion Is Not an Idempotent Operation

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Table of Contents
Part 1: Sets
Chapter 2: Constructions With Sets
Section 2.4: Powersets
Subsection 2.4.5: Powersets as Free Cocompletions
Warning 2.4.5.1.3: Free Cocompletion Is Not an Idempotent Operation (cite)

Warning 2.4.5.1.3 ▶ Free Cocompletion Is Not an Idempotent Operation

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Although the assignment $X\mapsto \mathcal{P}\webleft (X\webright )$ is called the free cocompletion of $X$, it is not an idempotent operation, i.e. we have $\mathcal{P}\webleft (\mathcal{P}\webleft (X\webright )\webright )\neq \mathcal{P}\webleft (X\webright )$.

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