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Mediation analysis is a powerful tool for studying causal pathways between exposure, mediator, and outcome variables of interest. While classical mediation analysis using observational data often requires strong and sometimes unrealistic assumptions, such as unconfoundedness, Mendelian Randomization (MR) avoids unmeasured confounding bias by employing genetic variants as instrumental variables. We develop a novel MR framework for mediation analysis with genome-wide associate study (GWAS) summary data, and provide solid statistical guarantees. Our framework efficiently integrates information stored in three independent GWAS summary data and mitigates the commonly encountered winner's curse and measurement error bias (a.k.a. instrument selection and weak instrument bias) in MR. As a result, our framework provides valid statistical inference for both direct and mediation effects with enhanced statistical efficiency. As part of this endeavor, we also demonstrate that the concept of winner's curse bias in mediation analysis with MR and summary data is more complex than previously documented in the classical two-sample MR literature, requiring special treatments to address such a bias issue. Through our theoretical investigations, we show that the proposed method delivers consistent and asymptotically normally distributed causal effect estimates. We illustrate the finite-sample performance of our approach through simulation experiments and a case study.