# Coproducts compute with finite limits in a reflective subcategoroy?

This question is related to this one: Which limits are preserved by a reflective left-adjoint?

(And in fact may be seen as a special case, but I think it merits its own question).

Suppose that $C$ is a reflective subcategory of $D$ and that coproducts in $D$ commute with finite limits. Notice that coproducts computed in $C$ are not necessarily the same as computing them in $D$; one may have to apply the reflector afterwards. Does it however follow that coproducts in $C$ commute with finite limits? Edit: No. See Todd's answer below. If not, under what assumptions will this be nonetheless true?

Edit: Obviously this is true if the reflector is left-exact. I am interested in conditions weaker than this.

Subquestion: Is there anything that can be said in the special situation where $C$ is the category of free-algebras for a monad on $D$ and the adjunction is the obvious one?

A simple example where finite limits do not commute with coproducts is where $D$ is the category of functors $T \to Set$ for just about any Lawvere theory $T$ that comes to mind, say the theory of groups, and $C$ is the category of product-preserving functors $T \to Set$. Clearly $C$ is a full subcategory of $D$ and the inclusion preserves limits and has a left adjoint, but in $C$ (which is equivalent to the category of groups), finite limits do not commute with coproducts.
• By the way, a notable exception to your family of counterexamples is the Lawvere theory of commutative $mathbb{R}$-algebras. Can we see what is special about this example at the level of Lawvere theories, that makes coproducts commute with finite limits? (By the way, my reflective subcategory is a subcategory of commutative $\mathbb{R}$-algebras as it turns out). – David Carchedi May 9 '12 at 2:18