Note: This is not intended to be a research level question, but concerns graduate level material.

**Theorem.** *The opposite $\Delta^\mathrm{op}$ of the simplex category $\Delta^\mathrm{op}$ (as usually defined via nondecreasing maps) can be presented as follows: it is the category generated by the graph of finite nonempty linearly ordered sets with arrows $\delta_i\colon [n]\to [n-1]$ and $\sigma_i\colon [n]\to [n+1]$ for each $0\leq i\leq n$ subject to the simplicial identities.*

Here is a proof sketch using the following well-known fact:

**Fact.** *Any nondecreasing map $f$ in $\Delta^\mathrm{op}$ has a unique representation as
$$f=\sigma^{i_1}\dots\sigma^{i_n}\delta^{j_1}\dots\delta^{j_m},$$
where $i_1>\dots > i_n$ and $j_1 <\dots < j_m.$*

Here, the $\sigma^i$ and $\delta^j$ are now regarded as nondecreasing maps, while in the above theorem they denote formal arrows. Sorry for the abuse of notation.

The above theorem means that for any category $\mathcal C$ there is a canonical bijection between functors $\Delta^\mathrm{op}\to\mathcal C$ and simplicial objects in $\mathcal C$ (given by face and degeneracy morphisms satisfying the simplicial identities). This is the universal property of being presented by generators and relations. By forgetting information one can a assign a simplicial object to each functor $\Delta^\mathrm{op}\to\mathcal C$. So the challenge is to construct an inverse to that assignment. Let $C_\bullet$ be a simplicial object with face maps $d_i$ and degeneracy maps $s_i$. We want to define a functor $\Delta^\mathrm{op}\to\mathcal C$. On objects this functor is given by $C_\bullet$. Now let $f\colon [n]\to [m]$. We need to find an induced morphism $$C(f)\colon C_n\to C_m.$$ To get one, write $$f=\sigma^{i_1}\dots\sigma^{i_n}\delta^{j_1}\dots\delta^{j_m},$$ and then set $$C(f)=s^{i_1}\dots s^{i_n}d^{j_1}\dots d^{j_m}.$$ One can now check that this defines a functor as desired.

*Question:* Is there a proof of the above theorem (i.e., a construction of the bijection between functors $\Delta^\mathrm{op}\to\mathcal C$ and simplicial objects of $\mathcal C$) that does *not* use the above fact? I wonder because the unique representation of a morphism $f$ in $\Delta^\mathrm{op}$ looks a bit random and ad hoc to me. In other settings when wanting to show that some concrete category can be presented by generators and relations, does one always convulsively have to find a way that each morphism can be written in a unique way as a composition of "generators" (satisfying additional properties, like $i_1 > \dots > i_n$ in the above case)?

I feel the theorem should be true nevertheless: just write $f$ as *any* composition of $\delta^i$ and $\sigma^i$s, and then map it to the same pattern where $\delta$ is replaced by $d$ and $\sigma$ is replaced by $s$. However, in this approach one has to check well-definedness. Can the well-definedness be checked without essentially reproducing the above fact?

The only advantage of the above fact seems to be to have a unique well-defined way to define the mapping. But I strongly feel like the theorem should be true even without particular ad hoc representation.

4more comments