  One often needs to turn a coupling $(X_i, Y_i)_{i\geq 0}$ of a Markov chain
into a sticky coupling where once $X_T = Y_T$ at some $T$, then from then on,
at each subsequent time step $T'\geq T$, we shall have $X_{T'} = Y_{T'}$.
However, not all of what are considered couplings in literature, even Markovian
couplings, can be turned into sticky couplings, as proved by Rosenthal through
a counter example. Rosenthal then proposed a strengthening of the Markovian
coupling notion, termed as faithful coupling, from which a sticky coupling can
indeed be obtained. We identify the reason why a sticky coupling could not be
obtained in the counter example of Rosenthal, which motivates us to define a
type of coupling which can obviously be turned into a sticky coupling. We show
then that the new type of coupling that we define, and the faithful coupling as
defined by Rosenthal, are actually identical. Our note may be seen as a
demonstration of the naturalness of the notion of faithful coupling.
