How to get a homomorphism of sheaves from a cohomomorphism?

Question

I have a problem in the book Sheaf Theory by Bredon.

Let $f:X\to Y$ be a continuous map of topological spaces. $\mathscr{A}$ and $\mathscr{B}$ are sheaves on $X$ and $Y$ respectively.

If $k:\mathscr{B}\to\mathscr{A}$ is an $f$-cohomomorphism, then we can obtain a function $h: f^*\mathscr{B}\to\mathscr{A}$.

I don't know how to check the continuity of $h$. Would you please give me some help?