or competing event (whichever comes first) and the corresponding event index that would be observed if the exposure $A$ were set to $a$ and the mediator levels changed to the levels that we would have seen if the exposure were set to $a^*$ and the levels of the time-varying confounders were as observed under this joint intervention on $A$ and $\overline{M}_K$, respectively. The total causal effect (TE) on the cause-$j$-specific hazard when the exposure changes from $a$ to $a^*$ is then expressed as $ HR^j_{TE} = \frac{\lambda^j_{a^*, a^*}(t)}{\lambda^j_{a, a}(t)}$, which can be decomposed into the direct effect (DE) $HR^j_{DE} = \frac{\lambda^j_{a^*, a}(t)}{\lambda^j_{a, a}(t)}$ and the indirect effect (IE) $HR^j_{IE} = \frac{\lambda^j_{a^*, a^*}(t)}{\lambda^j_{a^*, a}(t)}$, where $HR^j_{TE} = HR^j_{IE}\times HR^j_{DE}$. The indirect effect $HR^j_{IE} $ hence reflects the part of the treatment effect (on the cause-$j$-specific hazard) that is mediated via the pathways $A \rightarrow M_k \rightarrow \ldots \rightarrow T$, where $k = 1, 2,\ldots$. These pathways start from the treatment $A$ and go directly to one of the mediator levels before getting to the event time $T$ by any intermediate path. In contrast, the direct effect $HR^j_{DE} $ reflects the part of the treatment effect (on the cause-$j$-specific hazard) that does 