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