 Solution

The probability of a first event of type $A$ in $(u,u+du)$ is the probability $S(u)$ of still being at risk times $h_A(u)du$. Thus its cumulative incidence function isSimilarly $F_B(t)={\int }_0^t{h}_B(u)S(u)du$. In general $F_A(t)\ne 1-e^{-H_A(t)}$: that latter expression ignores the competing removal caused by $B$ and instead describes a net survival complement.