Unit-counit adjunction intuition Do you have a particular intuition or example you keep in your head when you think about the unit-counit adjunction? At this point, writing
$$R(\epsilon_c)\circ \eta_{R(c)} = \text{Id}_{R(c)}$$
$$\epsilon_{L(d)} \circ L(\eta_{d}) = \text{Id}_{L(d)}$$
just feels like symbolic manipulation to me. (Here we have $R:C \to D$ right adjoint to $L:D \to C$, and $c \in |C|$ and $d \in |D|$.)
Obviously there are many examples of adjoint functors. Looking for an clarifying and/or easy to remember example of how the unit and counit work together as above.
 A: 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.
A: Think in terms of identity.
Let me reference an object by two modalities, say |#> and |b>, with operators defined as


*

*|#|sharp

*|b| flat
and unit defined by 


*

*|#b| = 1 = |b#|


Now, to identify each object, let us create a round-trip path built form each objects unique spelling: we take a sharp path and immediately a flat path in the first object and we take a flat path and a immediately a sharp path in the second :


*

*|#> = |b#| |#> = 1 |#>

*|b> = |#b| |b> = 1 |b>.
Both of these are maps from the object back to itself, and thus should represent an identity, but we see that creating paths using our adjoint operators does not create a unique identity that separates objects created exclusively in one modality as opposed to the other.
But watch what happends when be bifurcate the original unit definition to include a counit


*

*|#b| = 1$_f$.

*|b#| = 1$_s$.
Now our objects unique round-trip path truly is unique for each object and can thus truly represent a unique identity:


*

*|#> = |b#| |#> = $1_s$ |#>

*|b> = |#b| |b> = $1_ f$ |b>.
(BTW, if I'm not mistaken, the first definition of our unit as self-dual corresponds to a monoidal interpretation for our space of study while our unit/counit definition corresponds to a group interpretation for our space)
