By the way, I found the following remark on [Milewski's blog](

> [T]he functor L is called the left adjoint to the functor R, while the functor R is the right adjoint to L. (Of course, left and right make sense only if you draw your diagrams one particular way.)

which seems to confirm that in this scenario the "left/right" direction has a **totally different** connotation from that in the finding-inverse-for-a-given-function problem, exactly as [Jonathan said above](

> I think there's a confusion between two very separate and distinct scenarios, where similar names are used for different things.

No idea if these two distinct scenarios for "left/right" have any deeper inner connection.