= Projectivity detected by all first left derived functors
{title2=$A\text{ projective}\iff L_1F(A)=0\text{ for every right exact }F$}
In an <abelian category> with enough projectives, a projective object has a resolution concentrated in degree zero, so every first <left derived functor> vanishes on it. Conversely choose $0\to K\to P\to A\to0$ with $P$ projective and use the additive <functor> $F=\mathcal A(-,K)$ with target the opposite of the <category> of <abelian groups>. This <functor> is <right exact> in that target. If $L_1F(A)=0$, the derived exact sequence makes $F(K)\to F(P)$ monic, meaning that $\operatorname{Hom}(P,K)\to\operatorname{Hom}(K,K)$ is surjective in <abelian groups>. Extending $1_K$ gives a retraction of $K\to P$, so the sequence splits and $A$ is a <direct summand> of the projective $P$.
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