Exercise 4.8.5.7. Let $f: X \rightarrow Y$ be a morphism of Kan complexes. Show that $f$ is full (in the sense of Definition 4.8.5.1) if and only if it satisfies the following pair of conditions:
- $(a)$
The map of connected components $\pi _0(f): \pi _0(X) \rightarrow \pi _0(Y)$ is injective.
- $(b)$
For every vertex $x \in X$ having image $y = f(x)$, the map of fundamental groups $\pi _{1}(f): \pi _1(X,x) \rightarrow \pi _{1}(Y,y)$ is surjective.