I tried formulating the universal property of the simplex category. There seemed to be two different options to choose for the universal property of $\Delta^{op}$. I'm having trouble making formulating of the one of them. I'm hoping to get some help with this, and maybe some input on which one is a better universal property to work with.
Here is my formulation of the universal property 1:
Let $S: Cat\to Set$ defined by: \begin{align*} S(C):=\{\{(C_n , \{d_i: C_n \to C_{n-1}\}_{0\leq i\leq n} , \{s_i : C_n\to C_{n+1}\}_{0\leq i\leq n})_{n\geq 0} \} : d_i, s_i \text{ satisfy the simplicial identities}\} \end{align*}
And, for any $F: C\to D$, define $F : S(C)\to S(D)$ by \begin{align*} \{(C_n , \{d_i\}_{0\leq i\leq n} , \{s_i \}_{0\leq i\leq n})_{n\geq 0} \} \mapsto \{(F(C_n) ,\{F(d_i)\}_{0\leq i\leq n} , \{F(s_i)\}_{0\leq i\leq n})_{n\geq 0} \} \end{align*}
since functorality preserves the simplicial identities.
CLAIM (UP 1) : $S$ is represented by $\Delta^{op}$, i.e, $Cat(\Delta^{op}, - ) \cong S$. Put differently, $(\Delta^{op} , \{([n], \{\delta_i\} , \sigma_i)\})$ is a universal arrow from $1\in Set$ to $S$. Here we think of $\{([n], \{\delta_i\} , \{\sigma_i\})\}$ as an arrow $1\to \Delta^{op}$ with image $\{([n], \{\delta_i\} , \{\sigma_i\})\}$. And $\delta_i$ and $\sigma_i$ are the notation for the face and degeneracy maps, respectively.
This UP1 seems to capture the idea that to specify a functor $\Delta^{op} \to C$, it suffices to specify an appropriate $(C_n , \{d_i\}_{0\leq i\leq n} , \{s_i \}_{0\leq i\leq n})_{n\geq 0}$
But, I often hear that $\Delta^{op}$ is freely generated by the $\delta_i$'s and the $\sigma_i$'s, subject to the simplicial identities. I am wondering how to best express this idea. It would be easier if the $\delta$'s and $\sigma$'s were not subject to any identities. Could this still be phrase in terms of the free category on a directed graph? There would need to be some sort of quotient happening. But I'm not sure how to express this in a free $\dashv$ forgetful type of universal property. Any help would be appreciated!