I'm having troubles with the exercise 1.1.1.10 on Kerodon. In particular, I don't understand how to prove the uniqueness of the sentence in italics; the paragraphs before it are mainly for context, so you can just skim through them.
Consider the set of natural numbers $\mathbb N$ as a totally ordered set, in the obvious way. Call $\Delta$ the category whose objects are the finite totally ordered subsets of $\mathbb N$, denoted as $[n]$ for $n\in \mathbb N$, with the injective order-preserving maps as morphisms. A semisimplicial object in (any category) $\sf C$ is a functor $\Delta^{op}\to\sf C$.
Define the maps $\delta_i:[n-1]\to [n]$, for $0\leq i\leq n$, by the rule: $\delta_i(j)=j$ if $j\lt i$, $\delta_i(j)=j+1$ if $j\ge i$. Notice that (in $\Delta$) any map $[n-1]\to [n]$ coincides with some $\delta_i$, and that the $\delta_i$ satisfy the property: $\delta_j\circ \delta _i=\delta_i\circ \delta _{j-1}$, as maps $[n-2]\to [n]$, for $0\leq i\lt j\leq n$.
Let $C:\Delta^{op}\to \sf C$ be a semisimplicial object. Denote $C([n])$ by $C_n$ and $C(\delta_i)$ by $d_n$. By functoriality the $d_i$ satisfy the property: ($*$) for $0\leq i\lt j\leq n$, $d_i\circ d _j=d_{j-1}\circ d _{i}$, as morphisms $C_n\to C_{n-2}$.
Also a converse holds: a family of objects $\{C_n\}_{n\in \mathbb N}$, plus $n+1$ morphisms $\{d_i:C_n\to C_{n-1}\}_{0\le i\le n}$ for all $n\ge1$, define a (unique) semisimplicial object $C$. Indeed in $\Delta$ any map $f:[m]\to [n]$, $m\lt n$, is uniquely determined by the elements (of $[n]$) not in the image of $f$, call them $i_1\lt\dots\lt i_{n-m}$, and $f$ is the composite $\delta_{i_{n-m}}\circ\dots\circ \delta_{i_{1}}$. Functoriality imposes to set $C(\delta_{i_{n-m}})\circ\dots\circ C(\delta_{i_{1}})$ as $C(f)$. However I don't understand why any $C(\delta_i)$ must be $d_i$, which I suppose is the unique choice meant in the italics. Probably one needs to use ($*$), since it has not been used yet, and I think that one could use induction some way, but I can't figure it out. Thanks in advance for any suggestion
$C(\delta_i)=d_i$ is not a claim. It is the meaning of "define" [a unique semisimplicial object]. I.e. the statement means: for every family of objects $\{C_n\}_{n\in \mathbb N}$ and $n+1$ morphisms $\{d_i:C_n\to C_{n-1}\}_{0\le i\le n}$ for all $n\ge1$ satisfying $(*)$, there is a unique functor $C:\Delta^{op}\to \sf C$ sending $[n]$ to $C_n$ ($\forall n\in\Bbb N$) and sending every $\delta^i$ to the corresponding $d_i$.
You just proved the uniqueness. For the existence, you need to use the hypothesis $(*).$