Zero surgery on a connected sum as a splice

Let $Y_1^{\prime }=X_1({\lambda }_1)$ and orient its filling core $K_1^{\prime }$ so that its longitude is ${\mu }_1$. Then zero surgery on $K_1\#K_2$ is the splice of $(Y_1^{\prime },K_1^{\prime })$ 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.