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Actually your guess is true. Consider the left Hopf algebra $\widetilde{SL_q(2)}$ from Iyer, Taft, The dual of a certain left quantum group. This is constructed essentially as $SL_q(2)$, but imposing …

The tensor product over a ring is not left exact Consider the short exact sequence $0\to2\mathbb{Z}\to\mathbb{Z}\to\mathbb{Z}_2\to0$ and tensor it by $\mathbb{Z}_2$ over $\mathbb{Z}$. It is well-know …

Not every epi is surjective Recall that if $\mathcal{C}$ is a category, then an arrow $p:x\to y$ in $\mathcal{C}$ is said to be an epi if for every pair of parallel arrows $f,g:y\to z$ such that $f\c …

Square root and square power are not inverse operations In fact, $\sqrt{x^2}=|{x}|$, which is defined for every $x\in\mathbb{R}$, while $({\sqrt{x}})^2=x$ is defined only for $x\geq 0$.

Is anybody aware of some more "concrete" or "natural" examples of this construction? This question is strictly related to this other question on MSE. Maybe that was not the right place where to ask. …