Let Prod(C, D) be the set of finite-product preserving functors from C to D. Is it true that for any Lawvere theory L, Prod(L, Set) has small colimits? This seems to be the case, as it is invoked here:


<http://ncatlab.org/nlab/show/database+of+categories>


Where do this colimits come from? In the case of sifted colimits, pointwise calculation works, as sifted colimits commute with finite products in Set. However, it doesn't seem at all obvious that an arbitrary colimit of product-preserving functors would also be product-preserving.


Does anyone know of a reference or proof that such colimits exist, or even better, how to go about computing them?



Computing colimits in a Lawvere theory
Let $T$ be a Lawvere theory. Then view the category of $T$-algebras, as Professor Blass wrote, as a class of universal algebras for a signature that corresponds to $T$ and is equationally definable, i.e. a variety. Then do what one does with these kinds of algebras:


To get coequalizers use quotients of algebras by what are called congruences (at least, that is what they are called in Universal Algebra). A congruence is an equivalence relation on the set that corresponds under your functor $A$ to the object 1 in $T$ (that is, the underlying set) that is preserved under all operations of the theory (arrows in $T$).


To get coproducts construct the left adjoint by defining the free $T$-algebra on some set $S$ to be the obvious structure on the set of all formal expressions $f(s\_1,...,s\_n)$ with $f$ an arrow of $T$ and $s\_1,...,s\_n\in S$. It is then clear what the coproduct of a set of free $T$-algebras is, and you can use congruences to get the coproduct on an arbitrary set of $T$-algebras.


