TITLE: Adjoint functors
QUESTION [7 upvotes]: I'm trying to wrap my brain around adjoint functors. One of the examples I've seen is the categories $\bf IntLE \bf = (\mathbb{Z}, ≤)$ and $\bf RealLE \bf = (\mathbb{R}, ≤)$, where the ceiling functor $ceil : \bf RealLE \rightarrow IntLE$ is left adjoint to the inclusion functor $incl : \bf IntLE \rightarrow RealLE$. I want to check that the following are true, as they seem to be:

$floor : \bf RealLE \rightarrow IntLE$ would be right adjoint to $incl$
Between the dual categories of $\bf IntGE \bf = (\mathbb{Z}, ≥)$ and $\bf RealGE \bf = (\mathbb{R}, ≥)$, $ceil$ would be right adjoint to $incl$
Between $\bf RealGE$ and $\bf IntGE$, $floor$ would be left adjoint to $incl$

Is my understanding correct on these points?

REPLY [2 votes]: Arturo has already posted a nice answer. I'd merely like to emphasize that such universal definitions often enable slick proofs, e.g. see below. For a much more striking example see the theorem in my post here, which presents a slick one-line proof of the  LCM * GCD law via their universal definitions.
LEMMA $\rm\: \  \lfloor x/(mn)\rfloor\ =\ \lfloor{\lfloor x/m\rfloor}/n\rfloor\ \  $
for $\rm\ \ n > 0$  
Proof $\rm\quad\quad\quad\quad\quad\quad\quad k\ \le \lfloor{\lfloor x/m\rfloor}/n\rfloor$
$\rm\quad\quad\quad\quad\quad\iff\quad\ \ k\ \le\ \:{\lfloor x/m\rfloor}/n$
$\rm\quad\quad\quad\quad\quad\iff\ \ nk\ \le\ \ \lfloor x/m\rfloor$
$\rm\quad\quad\quad\quad\quad\iff\ \ nk\ \le\:\ \ \ x/m$
$\rm\quad\quad\quad\quad\quad\iff\ \ \ \ k\ \le\:\ \ \ x/(mn)$
$\rm\quad\quad\quad\quad\quad\iff\ \ \ \  k\ \le\ \ \lfloor x/(mn)\rfloor $
Compare the above trivial proof to more traditional proofs, e.g. the special case $\rm\ m = 1\ $ here.