We investigate the relation between the {\em set-theoretical} description of
coinduction based on Tarski Fixpoint Theorem, and the {\em categorical} description of
coinduction based on {\em coalgebras}. In particular, we examine set-theoretic
generalizations of the {\em coinduction proof principle}, in the spirit of Milner's
{\em bisimulation ``up-to''}, and we discuss categorical counterparts for these. 
Moreover, we investigate the connection between these and the equivalences induced
by {\em $T$-coiterative functions}. These are morphisms into {\em final coalgebras},
satisfying the {\em $T$-coiteration scheme}, which is a generalization of both the
{\em coiteration } and the {\em corecursion  scheme}. We generalize Rutten's
transformation from coalgebraic bisimulations to set-theoretic bisimulations, in order
to cover also the case of bisimulations ``up-to''. A list of  examples of
set-theoretic coinductive specifications which appear not to be easily expressible in
coalgebraic terms are discussed.