In his book "Representations of semisimple Lie algebras in BGG category $\mathcal O$" (in this text I'm using notation from this book) J. Humphreys proves the following theorem (it's the proposition 1.13 in his book):
Let $\lambda$ be an integral weight for semisimple Lie algebra $\mathfrak g$ (of course, $\lambda \in \mathfrak h^*$) and let $\chi_\lambda$ denote corresponding central character $\chi_\lambda: Z(\mathfrak g) \to \mathbb C$. Then the subcategory $\mathcal O_{\chi_\lambda}$ of the category $\mathcal O$ (for Lie algebra $\mathfrak g$) is $\textit{a block}$.
Here is the proof which is given by Humphreys:
Let $M(\alpha)$ denote a Verma module associated with an arbitrary weight $\alpha$, let $N(\alpha)$ denote its unique maximal proper submodule and let $L(\alpha)$ denote unique simple quotient of $M(\alpha)$, i.e. $M(\alpha)/N(\alpha)$.
From the previous chapter, it follows that it just has to be shown that all modules $L(w \cdot \lambda)$ (for all possible $w$ in Weyl group of algebra $\mathfrak g$) lie in the same block. Using a simple induction argument we can see that it's enough to prove that for every integral weight $\lambda$ and for every simple reflection $s_\alpha$ modules $L(\lambda)$ and $L(s_\alpha \cdot \lambda)$ lie in the came block (Warning: here $s_\alpha \cdot \lambda$ means the so-called dot-product: $s_\alpha \cdot \lambda = s_\alpha(\lambda + \rho) - \rho$, where $\rho$ is the half-sum of all positive roots for adjoint action of Lie algebra $\mathfrak g$).
So we'll now prove that $L(\lambda)$ and $L(s_\alpha \cdot \lambda)$ are in the same block. Let $\mu$ denote $s_\alpha \cdot \lambda$. We can assume that $\mu \leq \lambda$ (if $\mu > \lambda$ we can reverse the roles of $\lambda$ and $\mu$). In the previous part of the book was proven the fact that in this case there is a non-zero homomorphism $\phi: M(\mu) \to M(\lambda)$ such that $N(\lambda)$ contains $\operatorname{Im}(\phi)$.
Let $N$ denote $\phi(N(\mu))$. Now we see that $\phi: M(\mu) \to M(\lambda)$ induces a morphism $\psi:L(\mu) = M(\mu) / N(\mu) \to M(\lambda)/N$. Moreover, $\operatorname{Ker}(\phi)$ is a proper submodule of $M(\mu)$ so $\operatorname{Ker}(\phi) \subset N(\mu)$. Thus $\psi$ is an embedding.
After that Humphreys says that from this fact it follows that $L(\mu)$ and $L(\lambda)$ are in the same block of category $\mathcal O$. Why? Can someone, please, explain to me this fact?
It would be clear if $L(\lambda)$ was isomorphic to $(M(\lambda)/N)/\psi(L(\lambda)) = (M(\lambda)/N)/\operatorname{Im}(\psi) \simeq M(\lambda)/\operatorname{Im}(\phi)$, but I don't see why this fact could be true...