This talk aims to answer two main questions: Can foundational theories be explanatory? If yes, can explanatoriness be used as a criterion for the preference of specific foundational theories? This is investigated by bringing together two debates, the one about explanations in mathematics and the other about criteria for preference of foundational theories. To answer the first question, an account of explanatoriness of foundational theories is developed. Here foundational theories are the explanans for mathematics (the explanandum) by fulfilling at least one of the criteria of Explanatory Systematization (ES) or Explanatory Reduction (ER), therefore also allowing for degrees of explanatoriness. ES can be expressed by epistemic accounts of explanation such as the unificatory account proposed by Kitcher (1989). ER corresponds to ontic accounts and is expressed here by the account of Pincock (2015) where explanation derives from dependency relations between objects of different abstractness. With this, the second questions is answered: It is argued that set theory fulfills ES but not ER, whereas category theory fulfills ER but not ES. Therefore different preferences for foundational theories can be linked to different ways in which these theories explain.