Let me try to spell out my understanding of the left/right directions (please correct me if I'm mistaken):

"Left" and "right" only make sense _with respect to_ something, and in Galois connections they are (meant to be) with respect to the **given function** (i.e. the function whose inverse/adjoint we are supposed to find), such that when we draw out the construction as [Michael Hong](https://forum.azimuthproject.org/discussion/comment/16766/#Comment_16766) does, an instance (arrow) of left adjoint is adjoined to the left of an instance of the given function, and an instance of right adjoint is adjoined to the right of an instance of the given function, both in the sense that the adjoint arrow's end point is connected to the given function arrow's start point.

A tricky issue here is that there is still another potential **reference point** in the construction for us to interpret the "left/right" direction, namely the (perhaps nonexistent) correct answer to the inverse-finding problem. A left adjoint arrow, involving the ceiling function, is to the right of the "correct answer", while a right adjoint arrow, involving the floor function, is to the left of the "correct answer".

In sum, if we change the reference point, the "left/right" relations also switch. So it's important to bear in mind that the "left/right" direction of an adjoint is with respect to the given function it's adjoined to.

It took me a while to figure this out, and perhaps it's only me, but in case anyone else is having the same difficulty understanding the definition of Galois connections, I hope these notes will be of some help. :-)

"Left" and "right" only make sense _with respect to_ something, and in Galois connections they are (meant to be) with respect to the **given function** (i.e. the function whose inverse/adjoint we are supposed to find), such that when we draw out the construction as [Michael Hong](https://forum.azimuthproject.org/discussion/comment/16766/#Comment_16766) does, an instance (arrow) of left adjoint is adjoined to the left of an instance of the given function, and an instance of right adjoint is adjoined to the right of an instance of the given function, both in the sense that the adjoint arrow's end point is connected to the given function arrow's start point.

A tricky issue here is that there is still another potential **reference point** in the construction for us to interpret the "left/right" direction, namely the (perhaps nonexistent) correct answer to the inverse-finding problem. A left adjoint arrow, involving the ceiling function, is to the right of the "correct answer", while a right adjoint arrow, involving the floor function, is to the left of the "correct answer".

In sum, if we change the reference point, the "left/right" relations also switch. So it's important to bear in mind that the "left/right" direction of an adjoint is with respect to the given function it's adjoined to.

It took me a while to figure this out, and perhaps it's only me, but in case anyone else is having the same difficulty understanding the definition of Galois connections, I hope these notes will be of some help. :-)