Variety as a reflective subcategory

Remarks

The algebra $A / \theta_A$ embeds into the product of the algebras $A / \theta$ over all such $\theta$, so it lies in $\mathcal{V}$ by closure under subalgebras and products. For $f \colon A \to B$ with $B \in \mathcal{V}$, the image $f(A)$ is a subalgebra of $B$, so $A / \ker f \cong f(A) \in \mathcal{V}$. Thus $\ker f \supseteq \theta_A$, and $f$ factors through $q_A$; the factorization is unique because $q_A$ is surjective.

Only closure under isomorphic copies, subalgebras and products is used, so the same holds for every quasivariety and every prevariety. Example: for the variety of abelian groups inside groups, $\theta_G$ is the congruence of the commutator subgroup $[G, G]$, and the reflection is the abelianization $G \to G / [G, G]$. Applied to the absolutely free term algebra on a set $X$, the reflection gives the free algebra of $\mathcal{V}$ on $X$.

Going deeper

For a student

The reflection $A \mapsto A / \theta_A$ is a universal construction, and $\theta_A$ is the congruence generated by the pairs $(s(\bar a), t(\bar a))$ for identities $s \approx t$ of $\mathcal{V}$ and tuples $\bar a$ of $A$. Since the reflector is a left adjoint, it preserves colimits: the free product in $\mathcal{V}$ of two members is the reflection of their free product in $\mathbf{Alg}(\sigma)$. Composing with the free functor of $\mathbf{Alg}(\sigma)$ recovers the free functor of $\mathcal{V}$. Reflective subcategories are closed under limits, which matches the closure of varieties under products and under equalizers, which are subalgebras.