Causal mediation effect decomposition when you have multiple mediators Suppose we have outcome variable $Y,$ one treatment $T,$ and two
mediator variables, $M_{1}$ and $M_{2}.$ We write the structural
model as:
$$
Y=\beta_{0}+\beta_{1}T+\beta_{2}M_{1}+\beta_{3}M_{2}+\epsilon
$$
$$
M_{1}=\alpha+\alpha_{1}T+v
$$
$$
M_{2}=\gamma+\gamma_{1}T+e
$$
Here, the direct effect of $T$ is straightforward, which is $\beta_{1}$.
Is the total effect $\beta_{1}+\beta_{2}\alpha_{1}+\gamma_{1}\beta_{3}?$
Also, is the causal medion effect is for $M_{1}$ given $\beta_{2}\alpha_{1}$
while that for $M_{2}$ given as $\gamma_{1}\beta_{3}?$ Does it make
sense to have two different mediation effects here?

Based on the model that you have specified you are correct

*

*Direct effect: $\beta_1$

*Indirect effect through $M_1$: $\alpha_1\cdot \beta_2$

*Indirect effect through $M_2$: $\gamma_1\cdot \beta_3$
It makes sense to decompose the indirect effect into a component that is due to each of the 2 mediators. The total indirect effect would be $\alpha_1\cdot \beta_2+\gamma_1\cdot \beta_3$
A detailed introduction to this decomposition can be found in Vanderweele 2014, which also describes how to incorporate a potential $M_1\cdot M_2$ interaction
