Example 8.2.6.3. Let $X$ be a Kan complex, let $\lambda _{-}: \operatorname{Fun}( \Delta ^1, X) \rightarrow X$ be the morphism given by evaluation at the vertex $0 \in \Delta ^1$, and let $\lambda _{+}: \operatorname{Fun}( \Delta ^1, X) \rightarrow X$ be the morphism given by evaluation at the vertex $1 \in \Delta ^1$. It follows from Corollary 3.1.3.3 that the map
is a Kan fibration; in particular, we can view it as a coupling of $X$ with itself. For each vertex $x \in X$, the path spaces $\lambda _{-}^{-1} \{ x\} = \{ x\} \operatorname{\vec{\times }}_{X} X$ and $\lambda _{+}^{-1} \{ x\} = X \operatorname{\vec{\times }}_{X} \{ x\} $ are contractible Kan complexes (Example 3.4.1.14), so that every object of $\operatorname{Fun}( \Delta ^1, X)$ is both universal and couniversal for the coupling $\lambda $. In particular, $\lambda $ is a balanced coupling.