Unit and counit adjunction

 Lecture topic

 Let $\mathcal F:C\to D$ and $\mathcal G:D\to C$ be adjoint functors. That is, let $\mathcal F$ be left-adjoint to $\mathcal G$, and let $\mathcal G$ be right-adjoint to $\mathcal F$, so that $\Hom_D(\mathcal F(X),Y)\cong\Hom_C(X,\mathcal G(Y))$ for any $X\in\Obj(C)$ and $Y\in \Obj(D)$.

This isomorphism gives natural maps $\eta_X$, from $$\begin{align*} \Hom_D(\mathcal F(X),\mathcal F(X)) & \cong \Hom_C(X,\mathcal G(\mathcal F(X)),\\ \id_{\mathcal F(X)} & \mapsto \left(X\tov{\eta_X}(\mathcal G\circ \mathcal F)(X)\right), \end{align*}$$ and $\epsilon_Y$, from $$\begin{align*} \Hom_C(\mathcal G(Y),\mathcal G(Y)) & \cong \Hom_D(\mathcal F(\mathcal G(Y)),Y), \\ \id_{\mathcal G(Y)} & \mapsto \left((\mathcal F\circ \mathcal G)(Y) \tov{\epsilon_Y}Y\right). \end{align*}$$ These may be viewed as natural transformations called the unit $\eta$ and the counit $\epsilon$, $$\eta:1_C \to \mathcal G\circ \mathcal F \hspace{2cm} \epsilon: \mathcal F\circ \mathcal G \to 1_D.$$ They satisfy the triangle identities, that is, the following diagrams commute.