Here is an example of the kind of result I have in mind:

>> **Proposition:** Let $X$ be a Noetherian scheme and $\mathcal C = \mathsf{QCoh}(X)$ its category of quasicoherent sheaves. Then $\mathcal C$ has enough projectives (if and) only if $X$ is affine.

More generally, if $\mathcal C$ is a locally finitely presentable abelian category which is "Noetherian" in the sense that any subobject of a compact object is compact (perhaps this can be relaxed slightly), then $\mathcal C$ has enough projectives iff it is an additive presheaf category. The modifications to the following proof necessary for this statement should be clear.

**Proof:** "If": in this case $\mathcal C$ is a module category.

"Only if": Choose (the zeroth step of) a projective resolution $P \twoheadrightarrow \mathcal O_X$. Present $P$ as a filtered colimit of coherent sheaves $P = \varinjlim_i C_i$. Because $X$ is Noetherian and $\mathcal O_X$ is coherent, there must be a $C = C_i$ such that the composite $C \to P \to \mathcal O_X$ is an epimorphism. Because $P$ is projective, we may choose a map $P \to C$ which commutes with the epimorphisms to $\mathcal O_X$. By composition, we obtain endomorphisms $e_C,e_P$ of $C,P$ which lie over the epimorphisms to $\mathcal O_X$, with commuting maps between them. For each $n \in \mathbb{N}$, the image $C^{(n)}$ of $e_C^n$ likewise maps epimorphically to $\mathcal O_X$ and hence is a generator. Because $X$ is Noetherian and $C$ is coherent, the $C^{(n)}$'s are coherent, and there is an $N$ such that $e_C$ restricts to an automorphism of $C^{(N)}$. Using the maps between $P$ and $C$, one can show that $C^{(N)} = P^{(N+1)}$ is a retract of $P$. So $C^{(N)}$ is a compact projective generator, so $\mathcal C$ is a module category, i.e. $X$ is affine.

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But ideally I would like a statement with looser hypotheses.