Difference between the characterizations of an $\infty$-category and nerves

Question

In Kerodon, Propposition 1.2.4.1 Proves that a simplicial set $S_\bullet$ is isomorphic to a nerve of an ordinary category if and only if every horn $\Lambda^n_i$ in $S_\bullet$, with $0<i<n$, has a filler. Then the same characterization is used to define $\infty$-categories (Definition 1.3.0.1). To me, this seems to imply that every $\infty$-category is isomorphic to a nerve of an ordinary category. But, the composition of morphism in a nerve is unique while in an $\infty$-category it is only unique up to homotopy, so being isomorphic to a nerve seems like a stronger condition. What am I missing here?

Answer 1

Read more carefully. The proposition states that horns can be filled uniquely. This is exactly the difference between categories and higher categories.