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 introduce 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 the {\em corecursion scheme}. 
We show how to describe coalgebraic {\em $F$-bisimulations} as set-theoretical
ones. A list of examples of set-theoretic coinductions which appear not to be 
easily amenable to coalgebraic terms are discussed.