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We introduce Nakayama functors for coalgebras and investigate their basicproperties. These functors are expressed by certain (co)ends as in the finitecase discussed by Fuchs, Schaumann, and Schweigert. This observation allows usto define Nakayama functors for Frobenius tensor categories in an intrinsicway. As applications, we establish the categorical Radford $S^4$-formula forFrobenius tensor categories and obtain some related results. These aregeneralizations of works of Etingof, Nikshych, and Ostrik on finite tensorcategories and some known facts on co-Frobenius Hopf algebras.

Nakayama functors for coalgebras and their applications to Frobenius tensor categories