Section 9 of CWM's chapter on limits beings by introducing the adjunctions $D\dashv U \dashv I$ where $D$ is the forgetful functor, $D$ equips sets with the discrete topology, and $I$ equips sets with the indiscrete topology
I'm trying to obtain intuition for why $D$ is left adjoint and $I$ is right adjoint, but I'm having a lot of trouble. Thinking in terms of reflections, my inuition for a left adjoint here is "the orthogonal projection of a particular set on $\mathsf{Top}$". I don't see why this should be the set with the discrete topology and not the indiscrete one. I don't have any co-intuition for co-reflections..
Added: I understand the adjunctions, which are just statements of the fact that any function from a discrete topological space is continuous and any function to an indiscrete topological space is continuous. The question is really about left and right adjoints, in particular, intuition for their "handedness".
You've got your (co)reflections backwards. These two adjunctions give us two subcategories of $\mathsf{Top}$, the discrete and indiscrete spaces, which are coreflective and reflective respectively, and which are both equivalent to $\mathsf{Set}$.
Note also that
the connected components functor provides an additional adjoint $\pi_0 \dashv D$,analogous "discrete $\dashv$ underlying $\dashv$ indiscrete" adjunctions between some category and $\mathsf{Set}$ show up in other places, cf. this comment by Steve Lack, and that often "connected components" provides a 4th adjoint in the string, $\pi_0 \dashv D$, but that $\pi_0 \not \dashv D$ in the case of $\mathsf{Top}$, although we do have $\pi_0 \dashv D$ if we replace $\mathsf{Top}$ by the subcategory of locally connected spaces.