  For any k-coalgebra C it is shown that similar quasi-finite C-comodules have
strongly equivalent coendomorphism coalgebras;
  (the converse is in general not true). As an application we give a general
result about codepth two coalgebra homomorphisms. Also a notion of codepth two
bicomodule is introduced. The last section applies similarity to an
endomorphism ring theorem for quasi-Frobenius (QF) bimodules and then to finite
depth ring extensions. For QF extensions, we establish that left and right
depth two are equivalent notions as well as a converse endomorphism theorem,
and characterize depth three in terms of separability and depth two.
