FALSE: every abelian category is equivalent to a module category

The category ${\mathbf{F}\mathbf{i}\mathbf{n}\mathbf{V}\mathbf{e}\mathbf{c}\mathbf{t}}_F$ is abelian: kernels, cokernels, images, coimages, and finite direct sums of linear maps between finite-dimensional vector spaces stay finite-dimensional, so the abelian-category structure of ${\mathbf{V}\mathbf{e}\mathbf{c}\mathbf{t}}_F$ restricts to this full subcategory.

The countable coproduct of countably many copies of the one-dimensional space $F$ does not exist in ${\mathbf{F}\mathbf{i}\mathbf{n}\mathbf{V}\mathbf{e}\mathbf{c}\mathbf{t}}_F$, because its usual direct sum is infinite-dimensional. But [L2] says every module category has all small coproducts. Since equivalences preserve which small coproducts exist, ${\mathbf{F}\mathbf{i}\mathbf{n}\mathbf{V}\mathbf{e}\mathbf{c}\mathbf{t}}_F$ cannot be equivalent to any module category. Therefore the universal statement is false.