# Proof forgetful functor $U:\mathsf{Mod}_\mathcal{T}\rightarrow \mathsf{Set}$ has a left adjoint

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.

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.