Adjoint Functor

A pair of functors $L:\mathcal{D}\to\mathcal{C}$ and $R:\mathcal{C}\to\mathcal{D}$ are adjoint, denoted $L\dashv R$, when

• There is a Natural Transformation $\eta:\mathrm{Id}\to R\circ L$, called the "unit" of the adjunction; and

• there is a Natural Transformation $\varepsilon:L\circ R\to\mathrm{Id}$, called the "counit" of the adjunction; and

• The zig-zag identities hold:

  	$\displaystyle\varepsilon\circ L(\eta)$	$\displaystyle=\mathrm{id}$
  	$\displaystyle R(\varepsilon)\circ\eta$	$\displaystyle=\mathrm{id}$