Corollary 4.7.7.2. Let $\kappa $ be an uncountable regular cardinal and let $f: X \rightarrow Y$ be a Kan fibration between Kan complexes, where $Y$ is essentially $\kappa $-small. The following conditions are equivalent:
The Kan complex $X$ is essentially $\kappa $-small.
For each vertex $y \in Y$, the fiber $X_{y} = \{ y\} \times _{Y} X$ is essentially $\kappa $-small.
Proof. The implication $(a) \Rightarrow (b)$ follows from Corollary 4.7.5.17 (and does not require the regularity of $\kappa $). Assume that condition $(b)$ is satisfied; we will show that $X$ satisfies the criteria of Proposition 4.7.7.1:
Let $y$ be a vertex of $Y$ and let $[y]$ denote its image in $\pi _0(Y)$. Since $f$ is a Kan fibration, the tautological map $\pi _0( X_ y ) \rightarrow \{ [y] \} \times _{ \pi _0(Y) } \pi _0(X)$ is a surjection. Assumption $(b)$ guarantees that $\pi _0(X_ y)$ is $\kappa $-small, so that the fiber $\{ [y] \} \times _{ \pi _0(Y) } \pi _0(X)$ is also $\kappa $-small. Since $\pi _0(Y)$ is $\kappa $-small, the regularity of $\kappa $ guarantees that $\pi _0(X)$ is also $\kappa $-small.
Fix a vertex $x \in X$ having image $y = f(x)$, and let $n > 0$ be a positive integer. For each integer $n > 0$, Proposition 3.2.6.2 supplies an exact sequence of groups
Consequently, every nonempty fiber of the group homomorphism $\pi _{n}(f)$ carries a transitive action of the $\kappa $-small group $\pi _{n}(X_ y, x)$, and is therefore $\kappa $-small. Since the group $\pi _{n}(Y,y)$ is $\kappa $-small, the regularity of $\kappa $ guarantees that $\pi _{n}(X,x)$ is $\kappa $-small.