Remark 4.3.5.16 (Slices of Coskeleta). Let $X$ and $Y$ be simplicial sets. For every integer $n \geq 0$, Remark 4.3.3.30 supplies a pullback diagram of sets
\[ \xymatrix { \operatorname{Hom}_{ \operatorname{Set_{\Delta }}}( \operatorname{sk}_{n}(Y^{\triangleright }), X) \ar [r] \ar [d] & \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \operatorname{sk}_{n}(Y), X) \ar [d] \\ \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \operatorname{sk}_{n-1}(Y)^{\triangleright }, X) \ar [r] & \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \operatorname{sk}_{n-1}(Y), X). } \]
Restricting the left side of the diagram to morphisms which carry the cone point of $Y^{\triangleright }$ to some fixed vertex $x \in x$ and invoking the universal properties of Example 4.3.5.15 and Remark 3.5.3.21, we obtain a pullback diagram of sets
\[ \xymatrix { \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( Y, \operatorname{cosk}_{n}(X)_{/x} ) \ar [r] \ar [d] & \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( Y, \operatorname{cosk}_{n}(X) ) \ar [d] \\ \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( Y, \operatorname{cosk}_{n-1}( X_{/x} ) ) \ar [r] & \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( Y, \operatorname{cosk}_{n-1}(X) ). } \]
This diagram depends functorially on $Y$, and therefore arises from a canonical isomorphism
\[ \operatorname{cosk}_{n}(X)_{/x} \xrightarrow {\sim } \operatorname{cosk}_{n-1}(X_{/x} ) \times _{ \operatorname{cosk}_{n-1}(X) } \operatorname{cosk}_{n}(X). \]