# ct.category theory – Define a sketch \$s_{mathbf{Grp}}\$ such that \$mathbf{Grp}backsimeq mathbf{Mod}(s_{mathbf{Grp}},mathbf{Set})\$

I have this MSE question with a two hundred bounty but even with the bounty this post got underviewed. So maybe here is a more suitable place to post it. The question 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 $$mathbf{Set}$$, use the result in (a) to prove that they also commute in $$mathbf{Grp}$$.

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

I found a useful example at nlab’s sketch article. Example 3.2 especifies the directed graph, diagrams, cones and cocones of a sketch which has unital magmas as models (sets with a binary operation which has a two sided unit).

So I thought taking this same sketch and “interpreting” the arrows $$e$$ as the identity of a group and $$m$$ as its multiplication, all this via the equivalence functor the exercise request us to build. But I don’t even know how to finish the construction of $$E$$ and in fact I don’t see why it will be an equivalence functor at all.

Could you help me? Also is there any result that I’m missing on (b)? Because I think this shouldn’t be so difficult.