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Zhen Lin
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In $\mathbf{Grp}$, the finitely presentable objects are precisely the finitely presented groups. Let $F_2$ be the free group on two elements. Then $F_2 \times F_2$ is finitely generated but not finitely presented, so the class of finitely presentable objects in $\mathbf{Grp}$ is not closed under binary products.

Zhen Lin
  • 15.9k
  • 1
  • 46
  • 84