FALSE: every abelian category has enough projectives and enough injectives

The category $\mathbf{F}\mathbf{i}\mathbf{n}\mathbf{A}\mathbf{b}$ is abelian. If it had enough projectives in the sense of [L1], then some nonzero projective object $P$ would surject onto $\mathbb{Z}\mathrm{/}p\mathbb{Z}$ for some prime $p$. Choose a cyclic quotient $u:P\twoheadrightarrow \mathbb{Z}\mathrm{/}{p}^m\mathbb{Z}$ with $m$ maximal among all cyclic $p$-power quotients of $P$.

The canonical quotient $q:\mathbb{Z}\mathrm{/}{p}^{m+1}\mathbb{Z}\twoheadrightarrow \mathbb{Z}\mathrm{/}{p}^m\mathbb{Z}$ is epic. If $P$ were projective, $u$ would lift across $q$, producing a surjection $P\twoheadrightarrow \mathbb{Z}\mathrm{/}{p}^{m+1}\mathbb{Z}$, contradicting maximality of $m$. So $\mathbf{F}\mathbf{i}\mathbf{n}\mathbf{A}\mathbf{b}$ does not have enough projectives, and the universal statement is false. The positive statements [L2] and [L3] are therefore extra-hypothesis results, not automatic consequences of abelianity.