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Self-supervised learning (SSL) recently has achieved outstanding success on recommendation. By setting up an auxiliary task (either predictive or contrastive), SSL can discover supervisory signals from the raw data without human annotation, which greatly mitigates the problem of sparse user-item interactions. However, most SSL-based recommendation models rely on general-purpose auxiliary tasks, e.g., maximizing correspondence between node representations learned from the original and perturbed interaction graphs, which are explicitly irrelevant to the recommendation task. Accordingly, the rich semantics reflected by social relationships and item categories, which lie in the recommendation data-based heterogeneous graphs, are not fully exploited. To explore recommendation-specific auxiliary tasks, we first quantitatively analyze the heterogeneous interaction data and find a strong positive correlation between the interactions and the number of user-item paths induced by meta-paths. Based on the finding, we design two auxiliary tasks that are tightly coupled with the target task (one is predictive and the other one is contrastive) towards connecting recommendation with the self-supervision signals hiding in the positive correlation. Finally, a model-agnostic DUal-Auxiliary Learning (DUAL) framework which unifies the SSL and recommendation tasks is developed. The extensive experiments conducted on three real-world datasets demonstrate that DUAL can significantly improve recommendation, reaching the state-of-the-art performance.