Isbell Duality

Mathematicians love dualities. After a brief explanation of dualities, with examples, we turn to one of the purest and most beautiful: Isbell duality. For any category $\mathsf{C}$, this gives an adjunction between the category of presheaves on $\mathsf{C}$, namely the functor category $[{\mathsf{C}}^{\text{op}},\mathsf{S}\mathsf{e}\mathsf{t}]$, and the opposite of the category of copresheaves on $\mathsf{C}$, namely $[\mathsf{C},\mathsf{S}\mathsf{e}\mathsf{t}{]}^{\text{op}}$.