# Relation of $ker\epsilon$ and $coker\eta$ in augmented algebra

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 $$ker(p)=(\eta:\mathbb{I} \to A)$$?