We will explain how Olivia Caramello's proposed approach for constructing a Galois-type Theory of cohomology functors can be implemented and fully verified in the case of cohomology of degree 0. The models of this Theory - or equivalently the points of the associated Classifying Topos - are exactly cohomology functors of degree 0. As its Classifying Topos is Galois, this Theory is complete, which means that all its models share the same geometric-logic properties. In particular, their components at all different geometric objects all have the same dimensions and the same algebraic structures. These results are already non-trivial and can be considered as toy-models for cohomology in higher degrees, which is the objective of Caramello's proposed approach.