On the proof of Weil conjectures in the curve case

I'm struggling to understand an argument in the book "Weil Conjectures, Perverse Sheaves, and $l$-adic Fourier Transform" by Kiehl and Weissauer.

In Theorem I.6.1, they prove (in specific cases) that the $i$-th cohomology of a pure sheaf of weight $w$ has weight $w+i$. I'm confused by their argument on the second half of page $47$ where they treat the special case of a geometrically constant sheaf.

Using the notation from the proof from now on: They want to show that that any eigenvalue $\alpha$ of Frobenius on $H_c^1(U,\mathscr{F})$ satisfies $|\tau(\alpha)|\le q^{w+1}$. They claim to deduce this by producing a surjection from $H^0(\mathbb{P}^1,\mathscr{H})$, and this cohomology group should have weights $\le w$, so a fortiori we seem to have proven the inequality we wanted.

So are we using some fact like "connecting homomorphisms in $l$-adic cohomology are frobenius equivariant"? This seems untrue, because then, if I'm not mistaken, their argument should prove the stronger (and false) claim that $H_c^1(U,\mathscr{F})$ has weight $w$ instead of $w+1$.

Perhaps my mistake came earlier, and I just don't understand this "semicontinuity of weights" theorem (I.2.8): Is it supposed to be the case that the weight of the sheaf $\mathscr{H}$ is bounded by $\le w+1$, and that it is not bounded by $\le w$? Such a statement now would seem to be compatible with the connecting homomorphism in etale cohomology being frobenius equivariant, but I fail to see how this follows from theorem I.2.8.