Hom from a projective counts simple composition factors

Given: The Axiom of Choice, a weight $\lambda$, the projective cover $P(\lambda )\twoheadrightarrow L(\lambda )$ of the previous theorem, and a finite-length object $X\in \mathcal{O}$.

$P(\lambda )$ is projective, is indecomposable with local endomorphism ring, has a unique maximal proper subobject $J(P(\lambda ))$ with $P(\lambda )\mathrm{/}J(P(\lambda ))$ simple, and the canonical epimorphism onto the head is essential with head $L(\lambda )$. The functor $\mathrm{Hom}(P(\lambda ),-)$ is exact (Category O has enough projectives, Projective covers in O are indecomposable and unique, Projective object characterisations).

For a simple object $L(\mu )$ of $\mathcal{O}$ one has ${\mathrm{Hom}}_{\mathcal{O}}(P(\lambda ),L(\mu ))\cong \mathbb{C}$ if $\mu =\lambda$ and $0$ if $\mu \mathrm{\ne }\lambda$: a nonzero morphism $P(\lambda )\to L(\mu )$ is an epimorphism, so $L(\mu )$ is the head of $P(\lambda )$ and $\mu =\lambda$ by [F1]; and for $\mu =\lambda$ every nonzero morphism has kernel a maximal proper subobject, hence equal to $J(P(\lambda ))$ by uniqueness, so all morphisms factor through the fixed quotient $L(\lambda )$. Each endomorphism of this highest-weight simple acts by a scalar on its one-dimensional highest line, which generates the module, so $\mathrm{End}(L(\lambda ))=\mathbb{C}$. Simple labels are distinct by The simple objects of O. [F1]

Every object of $\mathcal{O}$ has a finite composition series, and Jordan–Hölder makes its simple multiplicities independent of the series (Composition series and composition factors of an object, Every object of O has finite length, Jordan-Holder theorem in an abelian category). For $0\to A\to X\to B\to 0$, concatenate a composition series of $A$ with the inverse images of a composition series of $B$: the resulting series of $X$ has precisely their combined factors, proving additivity. The empty series of zero has all multiplicities zero.