Zero surgery on a connected sum as a splice
= Zero surgery on a connected sum as a splice
Let $Y_1'=X_1(\lambda_1)$ and orient its filling core $K_1'$ so that its longitude is $\mu_1$. Then zero surgery on $K_1\mathbin\#K_2$ is the splice of $(Y_1',K_1')$ and $(Y_2,K_2)$. Cutting the connected-sum exterior along its decomposing annulus and applying the <annulus-quotient model of Dehn filling> identifies $\lambda_1$ with $\lambda_2$ and $\mu_1$ with $\mu_2$, which is exactly the splice gluing.