This paper is a contribution to the {\em foundations}  of  {\em
coinductive types} and {\em coiterative functions}, in {\em
(Hyper)set-theoretical Categories}, in terms of {\em coalgebras}. 

We consider {\em atoms} as first class citizens. First of all, 
we give a {\em sharpening}, in the way of cardinality, of Aczel's 
Special Final Coalgebra Theorem, which allows for good estimates of the
cardinality of the final coalgebra. To these end, we introduce the notion 
of {\em $\kappa$-$Y$-uniform functor}, which subsumes Aczel's original 
notion. 
We give also an n-ary version of it, and we show that the resulting class
of functors is closed under many interesting operations used in Final 
Semantics. 
  

We  define also {\em
canonical} {\em wellfounded} versions of the final coalgebras of functors 
uniform on maps. This  leads to a {\em reduction} of {\em coiteration} to 
{\em ordinal induction}, giving a possible answer to a question raised by 
Moss and Danner.

Finally, we introduce a {\em generalization} of the notion of 
{\em $F$-bisimulation} inspired by Aczel's notion of {\em precongruence}, 
and we show that it allows to extend the  theory of {\em categorical 
bisimulations} also to functors {\em non}-weakly preserving pullbacks.  

Examples, non-examples, and open questions are frequent in the paper.