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I'm confused by a proof in volume II of Borceux.

First, two facts:

Proposition 3.3.3 $\;\;\;$ Let $\mathsf{T}$ be an algebraic theory. Consider the functor $$U:\mathsf{Mod}_\mathcal{T}\longrightarrow \mathsf{Set}$$ of evaluation at $T^1$. Then $U$ is representable and each finite set with $n$ elements admits $\mathcal{T}(T^n,-)$ as a reflection along $U$.

Theorem 3.4.5 $\;\;\;$ Let $\mathsf{T}$ be an algebraic theory. The category $\mathsf{Mod}_\mathcal{T}$ is reflective in the category $\mathsf{Fun}(\mathcal T,\mathsf{Set})$ and therefore it is both complete and cocomplete.

Now we have:

Corollary 3.7.8 $\;\;\;$ Let $\mathcal T$ be an algebraic theory. The forgetful functor $U:\mathsf{Mod}_\mathcal T\longrightarrow \mathsf{Set}$ has a left adjoint.

Proof $\;\;\;$ By 3.3.3 and 3.4.5 the singleton admits $\mathcal T(T^1,-)$ as a reflection along $U$ from which $\coprod _{x\in X}\mathcal T(T^1,-)$ is the reflection of the set $X$ along $U$; see 3.8.3, volume 1.

The problem is there is no "3.8.3, volume 1". I don't understand what allows to pass to infinite coproducts.

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Let $C$ be any category whatsoever, and let $c \in C$ be an object. I claim that the representable functor $\text{Hom}(c, -) : C \to \text{Set}$ has a left adjoint iff all coproducts $\bigsqcup_S c$ exist, and that when this is true the left adjoint sends a set $S$ to $\bigsqcup_S c$. Theorem 3.4.5 guarantees that the $C$ you care about is cocomplete, so in particular all of these coproducts exist.

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  • $\begingroup$ Where can I find proof of this theorem regarding existence of a left adjoint to the hom-functor? $\endgroup$ – user153312 Aug 24 '15 at 22:32
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    $\begingroup$ @Exterior: I don't know. It's not a very hard exercise to just do. $\endgroup$ – Qiaochu Yuan Aug 24 '15 at 22:33

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