Exercise 4.7.7.3. Let $\kappa $ be an uncountable regular cardinal and let $f: X \rightarrow Y$ be Kan fibration between Kan complexes. Suppose that $X$ is essentially $\kappa $-small, that each fiber $X_{y} = \{ y\} \times _{Y} X$ is essentially $\kappa $-small, and that the morphism $\pi _0(f): \pi _0(X) \rightarrow \pi _0(Y)$ is surjective. Show that $Y$ is also essentially $\kappa $-small.
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