Let $\mathcal{C}$ and $\mathcal{D}$ be two categories. Suppose that

- $F\colon\mathcal{C}\to\mathcal{D}$ is a functor from $\mathcal{C}$ to $\mathcal{D}$;
- $U\colon\mathcal{D}\to\mathcal{C}$ is a functor from $\mathcal{D}$ to $\mathcal{C}$;
- $F$ is left adjoint to $U$; and
- $\eta\colon\mathrm{id}_\mathcal{D}\to UF$ is the natural isomorphism forming the unit of the adjunction $(F, U, \eta)$.

Let $\epsilon$ be the inverse of $\eta$. My question is:Is $\epsilon$ **the** counit of the adjunction $(F, U, \eta)$ (which should now be a natural equivalence)?

notinverse for the first unit, even if both are isomorphisms. $\endgroup$ – მამუკა ჯიბლაძე Dec 28 '18 at 18:03