A limiting cone for a diagram is exactly a colimiting cocone for the formally dual diagram in the opposite category

Reversing ${\lambda }_j:L\to D(j)$ gives ${\lambda }_j^{\mathrm{o}\mathrm{p}}:D(j)\to L$. The cone equation $D(u){\lambda }_j={\lambda }_k$ reverses to the cocone equation ${\lambda }_j^{\mathrm{o}\mathrm{p}}D(u{)}^{\mathrm{o}\mathrm{p}}={\lambda }_k^{\mathrm{o}\mathrm{p}}$.

A cone morphism $h:(X,\xi )\to (L,\lambda )$ reverses to a cocone morphism from $(L,{\lambda }^{\mathrm{o}\mathrm{p}})$ to $(X,{\xi }^{\mathrm{o}\mathrm{p}})$, and this operation is bijective on morphisms.

Consequently terminality of $(L,\lambda )$ among cones is exactly initiality of $(L,{\lambda }^{\mathrm{o}\mathrm{p}})$ among cocones. By [F1], this proves both directions of the asserted equivalence.

Applying the same translation a second time gives the colimit-to-limit statement. Any later appeal to duality uses this exact reversal of objects, arrows, hypotheses, and conclusion, as required by [L1].