  A depth two extension $A \| B$ is shown to be weak depth two over its double
centralizer $V_A(V_A(B))$ if this is separable over $B$. We consider various
examples and non-examples of depth one and two properties. Depth two and its
relationship to direct and tensor product of algebras as well as cup product of
relative Hochschild cochains is examined. Section~6 introduces a notion of
codepth two coalgebra homomorphism $g: C \to D$, dual to a depth two algebra
homomorphism. It is shown that the endomorphism ring of bicomodule
endomorphisms $\End {}^DC^D$ forms a right bialgebroid over the centralizer
subalgebra $g^*: D^* \to C^*$ of the dual algebra $C^*$.
