Unit-counit adjunction intuition

String diagrams are amazing when it comes to working in a bicategory, in particular the 2-category $\mathsf{Cat}$. Basically objects are represented by regions of the plane, 1-morphisms are represented by string, and 2-morphisms are represented by nodes. Identities can be safely not represented on the diagram, and then the two triangle equalities simply become:

, and

The interesting thing, too, is that string diagrams are also used for monoidal categories (which are a special case of bicategories with a single object!). The two triangle equalities actually mean that the two "1-cells" (which are really objects of the monoidal category) are dual to each other, with evaluation $\eta$ and coevaluation $\epsilon$. I think it's an important point of view to regard adjoint functors as "dual" to each other in that sense.

If on the other hand you had started with the two-category $\mathsf{Top}$, then "adjointness" (in terms of triangle equalities) really means homotopy equivalence. So if you're comfortable with either duality or homotopy equivalence and how the different components of each one interact, you can use that to gain intuition about adjointness.

Explaining everything about string diagrams would be too long for a math.SE answer, but I hope the above gave you enough interest. There are a few references listed in the nLab article I linked at the beginning, and there's also an intro (specifically about 2-categories) in "Dualizability in Low-Dimensional Higher Category Theory" by Chris Schommer-Pries (in Topology and field theories, pp. 111–176, Contemp. Math., 613, Amer. Math. Soc., Providence, RI, 2014.), more specifically in section 6.