  It is well-known that the category of presheaf functors is complete and
cocomplete, and that the Yoneda embedding into the presheaf category preserves
products. However, the Yoneda embedding does not preserve coproducts. It is
perhaps less well-known that if we restrict the codomain of the Yoneda
embedding to the full subcategory of limit-preserving functors, then this
embedding preserves colimits, while still enjoying most of the other useful
properties of the Yoneda embedding. We call this modified embedding the Lambek
embedding. The category of limit-preserving functors is known to be a
reflective subcategory of the category of all functors, i.e., there is a left
adjoint for the inclusion functor. In the literature, the existence of this
left adjoint is often proved non-constructively, e.g., by an application of
Freyd's adjoint functor theorem. In this paper, we provide an alternative, more
constructive proof of this fact. We first explain the Lambek embedding and why
it preserves coproducts. Then we review some concepts from multi-sorted
algebras and observe that there is a one-to-one correspondence between
product-preserving presheaves and certain multi-sorted term algebras. We
provide a construction that freely turns any presheaf functor into a
product-preserving one, hence giving an explicit definition of the left adjoint
functor of the inclusion. Finally, we sketch how to extend our method to prove
that the subcategory of limit-preserving functors is also reflective.
