nLab coseparable coring

Coseparability of corings is a dual notion to separability of rings? (where one requires that the multiplication map is split).

An $A$-coring $(C,\Delta,\epsilon)$ is coseparable if the comultiplication $\Delta:C\to C\otimes_A C$ splits as a $C$-$C$-bicomodule morphism. In other words, there is a morphism of $A$-$A$-bimodules $p: C\otimes_A C\to C$ such that

$$\Delta\circ p = (C \otimes_A p)\circ(\Delta\otimes_A C) = (p\otimes_A C)\circ(C\otimes_A\Delta)$$

$$p\circ\Delta = C$$

where $C = \mathrm{Id}_C$.