We study \emph{observational equivalences} of programs 
of the object oriented language  \emph{Fickle}.
Fickle is a class-based  imperative language, which extends
Java with \emph{re-classification}. We introduce a \emph{coalgebraic
semantics} of Fickle programs, which is \emph{adequate} w.r.t. a
\emph{contextual} equivalence of programs, and \emph{compositional}
w.r.t. the operators of the language.  Moreover,
we discuss equivalences between programs implementing the same 
specification, expressed in terms of \emph{abstract classes}. 
We introduce a coalgebraic description of
classes, inspired by work of H.Reichel and B.Jacobs, which gives 
a coinductive equivalence on programs implementing the same specification. 
To this end, we extend the original coalgebraic  approach to \emph{binary 
methods}, i.e. methods
which take more than one instance of the hosting class as argument.
This coalgebraic description induces in particular a coinductive observational 
equivalence on
\emph{objects} of a program, where objects (states of a class) 
are taken to be equal when the
action of methods on them yields the same observations and equivalent next
states.