  For an effect algebra $A$, we examine the category of all morphisms from
finite Boolean algebras into $A$. This category can be described as a category
of elements of a presheaf $R(A)$ on the category of finite Boolean algebras. We
prove that some properties (being an orthoalgebra, the Riesz decomposition
property, being a Boolean algebra) of an effect algebra $A$ can be
characterized by properties of the category of elements of the presheaf $R(A)$.
We prove that the tensor product of of effect algebras arises as a left Kan
extension of the free product of finite Boolean algebras along the inclusion
functor. As a consequence, the tensor product of effect algebras can be
expressed by means of the Day convolution of presheaves on finite Boolean
algebras.
