Remark 4.3.1.9. The slice and coslice constructions of Construction 4.3.1.8 are mutually dual. More precisely, if $F: \operatorname{\mathcal{K}}\rightarrow \operatorname{\mathcal{C}}$ is a functor between categories and $F^{\operatorname{op}}: \operatorname{\mathcal{K}}^{\operatorname{op}} \rightarrow \operatorname{\mathcal{C}}^{\operatorname{op}}$ is the induced functor between opposite categories, then we have canonical isomorphisms
\[ (\operatorname{\mathcal{C}}_{/F})^{\operatorname{op}} \simeq (\operatorname{\mathcal{C}}^{\operatorname{op}})_{F^{\operatorname{op}}/} \quad \quad ( \operatorname{\mathcal{C}}_{F/} )^{\operatorname{op}} \simeq ( \operatorname{\mathcal{C}}^{\operatorname{op}} )_{ / F^{\operatorname{op}} }. \]