I am working my way through "Faisceaux pervers" by Beilinson, Bernstein and Deligne and can't wrap my head around Corollary 5.4.3. To me it seems that one of the hypotheses is extraneous, which surely must be wrong!
We are working with mixed $\overline{\mathbb{Q}}_l-$sheaves over schemes of finite type over $\mathbb{F}_q$. Corollary 5.4.2 states
If $f: X_0\to Y_0$ is a morphism of finite type, the functors $^pf_!$ and $^pf^*$ (resp. $^pf_*$ and $^pf^!$) preserve perverse sheaves of weight $\leqslant w$ (resp. of weight $\geqslant w$).
Corollary 5.4.3 then reads
If $f: X_0\to Y_0$ is a quasi-finite morphisms, $f_{!*}=\text{Im}(^pf_!\to\ ^pf_*)$ preserves perverse sheaves of weight $\leqslant w$ and of weight $\geqslant w$. In particular, it preserves perverse sheaves of pure weight $w$.
I am wondering whether the "quasi-finite" hypothesis is really needed. The proof I have in my mind is this: If $K_0$ is a perverse sheaf of weight $\leqslant w$, then $^pf_!K_0$ is also perverse of weight $\leqslant w$, and $f_{!*}K_0$ is perverse of weight $\leqslant w$ as its quotient. If $K_0$ is of weight $\geqslant w$, then $^pf_*K_0$ is of weight $\geqslant w$, and then $f_{!*}K_0$ is perverse of weight $\geqslant w$ as its subobject. When $K_0$ is pure, we just combine both the above cases.
Earlier, the same result was proven for $f$ affine and quasi-finite. The hypotheses were used to get that $f_!$ was $t$-exact with respect to the perverse $t$-structure, and the above argument with $^pf_!$ replaced with $f_!$ worked. However, I do not see why we need it any longer after Corollary 5.4.2, which shows that the perverse version of the image functors preserve weights (from one side).