Proposition 1.1.1.9. Let $\operatorname{\mathcal{C}}$ be a category and let $\{ C_ n \} _{n \geq 0}$ be a sequence of objects of $\operatorname{\mathcal{C}}$. Then a system of morphisms $\{ d^{n}_{i}: C_{n} \rightarrow C_{n-1} \} _{0 \leq i \leq n, n > 0}$ arise as the face operators of a semisimplicial object $C_{\bullet }$ of $\operatorname{\mathcal{C}}$ if and only if they satisfy condition $(\ast )$ of Remark 1.1.1.7. Moreover, if this condition is satisfied, then $C_{\bullet }$ is uniquely determined.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proof. Let $\widetilde{\operatorname{{\bf \Delta }}}_{\operatorname{inj}}$ denote the category which is freely generated by a collection of objects $\{ [n] \} _{n \geq 0}$ and a collection of morphisms $\{ \widetilde{\delta }_{n}^{i}: [n-1] \rightarrow [n] \} _{n > 0, 0 \leq i \leq n}$. Let $\overline{ \operatorname{{\bf \Delta }}}_{\operatorname{inj}}$ denote the quotient of $\widetilde{\operatorname{{\bf \Delta }}}_{\operatorname{inj}}$ obtained by imposing the relation
1.4
\begin{eqnarray} \label{equation:relation-semisimplicial-identity} \widetilde{\delta }^{j}_{n} \circ \widetilde{\delta }_{n-1}^{i} & = & \widetilde{\delta }_{n}^{i} \circ \widetilde{\delta }_{n-1}^{j-1} \end{eqnarray}
for every integer $n \geq 2$ and every pair $0 \leq i < j \leq n$. Using Remark 1.1.1.7, we see that there is a unique functor $F_{\operatorname{inj}}: \overline{\operatorname{{\bf \Delta }}}_{\operatorname{inj}} \rightarrow \operatorname{{\bf \Delta }}_{\operatorname{inj}}$ which carries each object $[n] \in \overline{\operatorname{{\bf \Delta }}}_{\operatorname{inj}}$ to itself, and each generating morphism $\widetilde{\delta }_{n}^{i}$ to the monomorphism $\delta _{n}^{i}: [n-1] \hookrightarrow [n]$ of Construction 1.1.1.4. To prove Proposition 1.1.1.9, it will suffice to show that the functor $F_{\operatorname{inj}}$ is an isomorphism of categories.
Fix integers $0 \leq m \leq n$, and set $b = n-m-1$. In the category $\widetilde{\operatorname{{\bf \Delta }}}_{\operatorname{inj}}$, every morphism $\beta : [m] \rightarrow [n]$ admits a unique factorization $\beta = \widetilde{\delta }_{n}^{i_0} \circ \widetilde{\delta }_{n-1}^{i_1} \circ \cdots \circ \widetilde{\delta }_{n-b}^{i_{b}}$, where the superscripts are nonnegative integers satisfying $0 \leq i_ a \leq n - a$ for $0 \leq a \leq b$. Let us say that $\beta $ is in standard form if, in addition, the integers $i_ a$ satisfy the inequalities $i_0 > i_1 > i_2 > \cdots > i_ b$. Note that, by repeatedly applying the relation (1.4), we can convert any morphism of $\widetilde{\operatorname{{\bf \Delta }}}_{\operatorname{inj}}$ to a morphism which is in standard form. More precisely, every morphism $\overline{\beta }: [m] \rightarrow [n]$ in $\overline{\operatorname{{\bf \Delta }}}_{\operatorname{inj}}$ can be lifted to a morphism $\beta : [m] \rightarrow [n]$ which is in standard form.
By construction, the functor $F_{\operatorname{inj}}$ is bijective on objects. To complete the proof, it will suffice to show that for every morphism $\alpha : [m] \hookrightarrow [n]$, there is a unique morphism $\overline{\beta }: [m] \rightarrow [n]$ in $\overline{\operatorname{{\bf \Delta }}}_{\operatorname{inj}}$ satisfying $F_{\operatorname{inj}}( \overline{\beta } ) = \alpha $. By virtue of the preceding discussion, it will suffice to show that $\alpha $ can be lifted uniquely to a morphism $\beta : [m] \rightarrow [n]$ in the category $\widetilde{\operatorname{{\bf \Delta }}}_{\operatorname{inj}}$ which is in standard form. We now observe that $\beta = \widetilde{\delta }_{n}^{i_0} \circ \widetilde{\delta }_{n-1}^{i_1} \circ \cdots \circ \widetilde{\delta }_{n-b}^{i_{b}}$ is characterized by the requirement that $\{ i_ b < i_{b-1} < \cdots < i_0 \} \subseteq [n]$ is the complement of the image of $\alpha $. $\square$