Small projective generators and progenerators

Let $\mathcal{C}$ be a locally small cocomplete abelian category. An object $P$ of $\mathcal{C}$ is a small projective generator when (i) $P$ is projective (Projective object), (ii) $P$ is a generator (Generator and cogenerator of a category), and (iii) the abelian-group-valued functor $\mathcal{C}(P,-):\mathcal{C}\to \mathbf{A}\mathbf{b}$ preserves every set-indexed coproduct (Preservation, reflection, and creation of limits and colimits; continuous and cocontinuous functors, Products and coproducts as limits and colimits of discrete diagrams, including their existence-and-uniqueness equations). Here "small" names this compactness property of the functor $\mathcal{C}(P,-)$; it does not assert that the underlying object, set, or module is small. For a unital ring $B$, a left $B$-module $P$ is a progenerator when $P$ is finitely generated (Generated submodule, cyclic and finitely generated modules, module basis and free module), projective (Projective modules and the lifting property), and a generator. For module categories the two notions agree: a left $B$-module is a small projective generator of $B\text{-Mod}$ if and only if it is a progenerator (Small projective modules are exactly finitely generated projective modules; the progenerator identification ↗), and in particular the regular module ${}_{B}B$ is a small projective generator. No commutativity of $B$ is assumed, and both phrases are properties of an object, not existence axioms beyond the coproducts already required of $\mathcal{C}$.