Sweedler notation and String diagrams: coaction on quotient

Question

Let $(C, \Delta, \epsilon)$ be a coalgebra (in the category $Vect$), and $I \subset C$ a right coideal, i.e. $\Delta (I) \subset I \otimes C$.

Define a counital coaction $\overline \Delta := (x_{(1)} + I) \otimes x_{(2)}$ of $C$ on $C/I$.

1. How would you write down $\overline \Delta$ as a composition of maps? In the following way: $(\pi \otimes id_C) \circ \Delta \circ \pi ^{-1} = \overline \Delta$, where $\pi: C \rightarrow C/I$ denotes the canonical projection and $\pi^{-1}$ a right inverse of $\pi$? The map $\pi^{-1}$ might not be linear, so that possibly $\overline \Delta$ is not linear?

EDIT (my own answer): The right inverse is not unique, one has to choose an inverse using the axiom of choice. One can always choose the right inverse to be linear. This is proven here: Linearity of the right inverse of a surjective linear map

2. How would you write down $\overline \Delta$ in a string diagram?
   In the following way? Here the map $\pi^{-1}$ denotes a linear right inverse of $\pi$ which always exists (see above). Hopefully, it is clear what convention in drawing the diagram I am using.