For a functor $S: \mathbf{A} \to \mathbf{B}$, if there is a functor $T: \mathbf{B} \to \mathbf{A}$ such that $TS \cong 1_{\mathbf{A}}$, then $S$ is faithful.
Can someone provide me an example of a faithful functor $S$ for which there does not exists a $T$ satisfying $TS \cong 1_{\mathbf{A}}$?
Simple, but important, examples of categories are ordered sets. They are sufficiently different than discrete categories (i.e., sets) to provide for counter examples. So, searching for a counter example in the form of two categories which are in fact ordered sets, note first that functors are just monotone mappings, and every functor is trivially faithful. So, you are merely looking for a monotone mappings with no left inverse. Now, a monotone mappings from a trivially ordered set is nothing but a function of the underlying sets. To construct such a function that fails to have a left inverse, simply consider a non-trivial codomain.