Unit and counit

Remarks

Both families are natural transformations, $\eta \colon \mathrm{Id}_{\mathcal{C}} \Rightarrow G F$ and $\varepsilon \colon F G \Rightarrow \mathrm{Id}_{\mathcal{D}}$. They satisfy the triangle identities $$\varepsilon_{F c} \circ F(\eta_c) = \id_{F c}, \qquad G(\varepsilon_d) \circ \eta_{G d} = \id_{G d}.$$ They recover $\varphi$ by $\varphi(g) = G(g) \circ \eta_c$ for $g \colon F c \to d$.

Example: for the free vector space functor $F$ over a field $k$ and the forgetful functor $G$, the unit $\eta_X \colon X \to G F X$ sends $x$ to the basis vector $e_x$. The counit $\varepsilon_V \colon F G V \to V$ sends the basis vector $e_v$ to the vector $v$. Then $\varepsilon_{F X}(F(\eta_X)(e_x)) = \varepsilon_{F X}(e_{e_x}) = e_x$, which checks the first triangle identity.

Non-example: if the characteristic of $k$ is not $2$, the natural maps $\eta'_X(x) = 2 e_x$ are not a unit for this counit. Here $\varepsilon_{F X} \circ F(\eta'_X)$ is multiplication by $2$, so the first triangle identity fails.