Cone of an adjunction I came across this sentence
"...let $\varepsilon: GG^\vee \to \operatorname{Id}$ be the counit of adjunction and $Z$ its cone."
I thought that cones were constructions on functors. $\varepsilon$, though, is a morphism of functors (a natural transformation)... in this context, what does "its cone" mean?
Or is the author talking about "the cone of an adjunction"? I thought that could be the case: if we take $G,G^\vee:C\to C$ adjoint functors, and construct a cone on $G^\vee$ $$N, \phi_X:N \to G^\vee X \quad \tiny{(\text{what object $N$ do we choose?})}$$ 
then I guess I could get a cone on $G: C \to C$ too by choosing $\tilde{N} = GN$ and defining $$\psi_X : GN \to X$$ getting $\psi_X=G(\phi_X)$ via $G$ functoriality and then  translating $GG^\vee X$ into $\operatorname{Id}_X$ with $\varepsilon$. So in some sense this could be "a cone of an adjunction".
Is this the right approach? If yes, is there a special $N$ to choose?

It means the mapping cone, in the sense of triangulated categories. A triangulated category is a category $\mathcal{C}$ equipped with an endofunctor $\Sigma : \mathcal{C} \to \mathcal{C}$ and a collection of distinguished triangles, each of which is a diagram of the form $X \to Y \to Z \to \Sigma X$. This data is required to satisfy several conditions, one of which is that every morphism $f : X \to Y$ is a part of some distinguished triangle $X \xrightarrow{f} Y \to Z \to \Sigma X$; the term cone refers to any of the possible $Z$'s.
