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Let $\mathbb{T}$ be a finitary algebraic theory and $\mathbb{T}\text{-Alg(Set)}$ be the category of finite-product-preserving functors $\mathbb{T} \rightarrow \text{Set}$.

It is written in my textbook that "It is known that the category $\mathbb{T}\text{-Alg(Set)}$ is complete and cocomplete."

I want to prove this. I can prove the completeness, since its limits are pointwise limit. However, the coproduct isn't pointwise coproduct.

How do you prove the cocompleteness?

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    $\begingroup$ It suffices to construct coproducts and coequalizers, which can both more or less be done by hand, taking inspiration from the case of groups. Once you know how to construct algebras via generators and relations, coequalizers are just imposing relations, and the coproduct of $A$ and $B$ is generated by every element of $A$ and every element of $B$ modulo every relation holding in $A$ and every relation holding in $B$. $\endgroup$ – Qiaochu Yuan Jul 16 '16 at 6:46
  • $\begingroup$ Just a question, by a finitary algebraic theory, do you mean as in Borceux's book a category $\mathbb{T} = \{T_0, T_1,...\}$ where each object $T_n$ is the $n$-th power of $T_1$? Or do you mean a variety of algebras? $\endgroup$ – Mike Jul 16 '16 at 13:17
  • $\begingroup$ I mean the former. $\endgroup$ – user53216 Jul 16 '16 at 13:29

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