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ct.class concept – Define a sketch $s_{mathbf{Grp}}$ such that $mathbf{Grp}backsimeq mathbf{Mod}(s_{mathbf{Grp}},mathbf{Set})$

I’ve this MSE query with a 200 bounty however plane with the bounty this put up obtained underviewed. So perhaps here’s a extra apt place to put up it. The query follows:

(a) Define a sketch $s_{mathbf{Grp}}$ and a equivalence functor $$E: mathbf{Grp}to mathbf{Mod}(s_{mathbf{Grp}},mathbf{Set})$$ (b) Knowing that finite limits commute with filtered colimits in $mathbf{Set}$, employ the outcome in (a) to show that they too commute in $mathbf{Grp}$.

(c) Prove that $mathbf{Ab} backsimeq mathbf{Mod}(s_{mathbf{Grp}},mathbf{Grp})$

I create a helpful instance at nlab’s sketch article. Example 3.2 especifies the directed graph, diagrams, cones and cocones of a sketch which has unital magmas as fashions (units with a binary operation which has a two sided unit).

So I assumed taking this identical sketch and “interpreting” the arrows $e$ because the identification of a gaggle and $m$ as its multiplication, all this through the equivalence functor the rehearse request us to construct. But I do not plane know the way to terminate the development of $E$ and in reality I do not behold why it should breathe an equivalence functor in any respect.

Could you ameliorate me? Also is there any outcome that I’m lacking on (b)? Because I cerebrate this should not breathe so tough.

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