Let $\mu,\lambda\in\mathfrak{h}^*$ with $\mu-\lambda\in \Lambda$ and let $pr_\mu$ and $pr_\lambda$ be natural projections of category $O$ onto blocks $O_{\chi_\mu}$ and $O_{\chi_\lambda}$ respectively. Now the $W-$orbit of $\gamma:=\mu-\lambda$ contains a unique weight $\bar{\gamma}\in \Lambda^+$. Set $L:=L(\bar{\gamma})$ and let $T_\lambda^\mu$ be the translation functor be as follows:
For $M\in O$, $T_\lambda^\mu$ sends $M$ to $pr_\mu(L\otimes(pr_\lambda M))$, followed by inclusion into O.
My question is: How to show that $Hom_O(T_\lambda^\mu M,N)\simeq Hom_O(M,T_\mu^\lambda N)$?
I can prove that $T_\mu^\lambda N=pr_\lambda(L^*\otimes(pr_\mu N))$ where $L$ is the dual of $L$.
I think I need to use the isomorphism $Hom_O(L\otimes M,N)\simeq Hom_O(M, L^*\otimes N)$. But I do not know how to deal with the projection part.
Because there are no non-zero homomorphisms between modules in distinct blocks and all the modules $M$ have the projection $pr_\mu M$ as a direct summand, we also have the following isomorphisms for all $\mu, M,N$: $$ Hom(pr_\mu M,N)\simeq Hom(pr_\mu M,pr_\mu N)\simeq Hom(M,pr_\mu N). $$ Using these in addition to the natural isomorphism you described it becomes simple: $$ \begin{aligned} Hom(T^\mu_\lambda M,N)&=Hom(pr_\mu(L\otimes pr_\lambda M),N)\\ &\simeq Hom(pr_\mu(L\otimes pr_\lambda M),pr_\mu N)\\ &\simeq Hom(L\otimes pr_\lambda M, pr_\mu N)\\ &\simeq Hom(pr_\lambda M, L^*\otimes pr_\mu N)\\ &\simeq Hom(pr_\lambda M, pr_\lambda(L^*\otimes pr_\mu N))\\ &\simeq Hom(M,pr_\lambda(L^*\otimes pr_\mu N))\\ &= Hom(M,T^\lambda_\mu N). \end{aligned} $$