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Let $C$ be a monoidal abelian category and $(A,\epsilon)$ an augmented algebra in $C$. Set $p:=id-\eta\circ \epsilon:A \to A$ (with $\eta$ being the unit).

  1. Is $coker(p)\cong(\epsilon:A \to \mathbb{I})$?
  2. Is $ker(\epsilon)=coker(\eta)$?
  3. Is $ker(p)=(\eta:\mathbb{I} \to A)$?
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