Let $\mathscr{C}$ be a category, $\mathscr{C}^{\mathrm{o}}$ be its opposite category and let $\mathscr{C}^{\vee}$ be the category of contravariant functors from $\mathscr{C}$ to $\mathbf{Set}$. The following image is from page 26 of the book Sheaves on Manifolds by Kashiwara and Schapira. Does it contain an error? It says $\mathscr{C}^{\vee\mathrm{o}}$ is (equivalent to) the category of covariant functors from $\mathscr{C}$ to $\mathbf{Set}$, denoted by $\mathscr{C}^{\wedge}$. Then it says $\mathscr{C}^{\wedge}\simeq\mathscr{C}^{\mathrm{o}\vee\mathrm{o}}$.
If this is true then one can conclude that $\mathscr{C}^{\vee\mathrm{o}}\simeq \mathscr{C}^{\mathrm{o}\vee\mathrm{o}}$.
I think the confusion comes from the fact that in the book Categories and Sheaves by the same authors, $\mathscr{C}^\wedge$ is defined as $$\mathscr{C}^\wedge:=[\mathscr{C},\mathbf{Set}]^{\rm{op}}$$ and not as the opposite category of $\mathscr C^{\vee}$, which would be $[\mathscr{C}^{\rm{op}},\mathbf{Set}]^{\rm{op}}\cong[\mathscr{C},\bf{Set}^{\rm{op}}]$ (actually I fail to see how this is the category of covariant functors into the category of sets). Using the above definition, the claimed isomorphism is obviously true.