Example 3.2.2.18 (Independence of Base Point). Let $X$ be a Kan complex and let $e: x \rightarrow y$ be an edge of $X$. Then evaluation at the vertices $0,1 \in \Delta ^1$ determines a diagram of pointed Kan complexes $(X,x) \xleftarrow { \operatorname{ev}_0 } ( \operatorname{Fun}( \Delta ^1, X), e) \xrightarrow { \operatorname{ev}_1 } (X,y)$, where the underlying maps are trivial Kan fibrations (Corollary 3.1.3.6). Applying For each $n > 0$, Remark 3.2.2.17 then supplies isomorphisms of homotopy groups
\[ \pi _{n}(X,x) \xleftarrow {\sim } \pi _{n}( \operatorname{Fun}(\Delta ^1, X), e) \xrightarrow {\sim } \pi _{n}(X,y). \]