Kerodon

Remark 3.2.2.16 (Functoriality). Let $f: X \rightarrow Y$ be a morphism of Kan complexes, let $x$ be a vertex of $X$, and set $y = f(x)$. For each $n \geq 1$, the morphism $f$ induces a homomorphism $\pi _{n}(f): \pi _{n}(X,x) \rightarrow \pi _{n}(Y,y)$, characterized by the formula $\pi _{n}(f)( [\sigma ] ) = [ f(\sigma ) ]$ for each $n$-simplex $\sigma $ of $X$ for which $\sigma |_{ \operatorname{\partial \Delta }^ n}$ is the constant map $\operatorname{\partial \Delta }^ n \rightarrow \{ x\} \hookrightarrow X$. We can therefore regard the construction $(X,x) \mapsto \pi _{n}(X,x)$ as a functor from the category of pointed Kan complexes to the category of groups. Moreover, this functor preserves filtered colimits.