Kerodon

Proof of Proposition 5.3.3.24. The construction $\operatorname{\mathcal{E}}\mapsto T_{\operatorname{\mathcal{E}}}$ carries colimits in the category $(\operatorname{Set_{\Delta }})_{ / \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) }$ to limits in the arrow category $\operatorname{Fun}( [1], \operatorname{Set})$. We can therefore assume without loss of generality that $\operatorname{\mathcal{E}}= \Delta ^ n$ is a standard simplex, so that the morphism $U$ determines a diagram $C_0 \rightarrow C_1 \rightarrow \cdots \rightarrow C_ n$ in the category $\operatorname{\mathcal{C}}$. Unwinding the definitions, we see that the codomain of $T_{\operatorname{\mathcal{E}}}$ can be identified with the set of tuples $\tau = (\tau _0, \tau _1, \cdots , \tau _ n)$, where $\tau _ i: \Delta ^{i} \rightarrow \mathscr {F}(C_ i)$ are simplices for which the diagram

is commutative. Let us regard $\tau $ as fixed; we wish to prove that there is a unique natural transformation $\alpha : \mathscr {G}_{\operatorname{\mathcal{E}}} \rightarrow \mathscr {F}$ satisfying $T_{\operatorname{\mathcal{E}}}(\alpha ) = \tau $.

Let $D$ be an object of $\operatorname{\mathcal{C}}$ and let $m \geq 0$ be an integer. Then $m$-simplices of the simplicial set $\mathscr {G}_{\operatorname{\mathcal{E}}}(D) = \operatorname{N}_{\bullet }( \operatorname{\mathcal{C}}_{/D} ) \times _{ \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) } \operatorname{\mathcal{E}}$ can be identified with pairs $(f, g)$, where $g: [m] \rightarrow [n]$ is a nondecreasing function and $f: C_{ g(m) } \rightarrow D$ is a morphism in the category $\operatorname{\mathcal{C}}$. Let $\alpha _{D}(f,g)$ denote the $m$-simplex of $\mathscr {F}(D)$ given by the composition

The construction $(f,g) \mapsto \alpha _{D}(f,g)$ determines a morphism of simplicial sets $\alpha _{D}: \mathscr {G}_{\operatorname{\mathcal{E}}}(D) \rightarrow \mathscr {F}(D)$. The assignment $D \mapsto \alpha _{D}$ determines a natural transformation of functors $\alpha : \mathscr {G}_{\operatorname{\mathcal{E}}} \rightarrow \mathscr {F}$ satisfying $T_{\operatorname{\mathcal{E}}}(\alpha ) = \tau $. This proves existence.

We now prove uniqueness. Suppose we are given another natural transformation $\alpha ': \mathscr {G}_{\operatorname{\mathcal{E}}} \rightarrow \mathscr {F}$ satisfying $T_{\operatorname{\mathcal{E}}}(\alpha ') = \tau $; we wish to show that $\alpha = \alpha '$. Fix an object $D \in \operatorname{\mathcal{C}}$ and an $m$-simplex of the simplicial set $\mathscr {G}_{\operatorname{\mathcal{E}}}(D)$, which we identify with a pair $(f,g)$ as above. We wish to verify that $\alpha _{D}(f,g)$ and $\alpha '_{D}(f,g)$ coincide (as $m$-simplices of the simplicial set $\mathscr {F}(D)$). Set $n' = g(m)$, so that the function $g$ factors as a composition $[m] \xrightarrow {g} [n'] \xrightarrow {\iota } [n]$, where $\iota : [n'] \hookrightarrow [n]$ is the inclusion map. Since $\alpha _ D$ and $\alpha '_{D}$ are morphisms of simplicial sets, it will suffice to prove that $\alpha _{D}( f, \iota )$ and $\alpha '_{D}(f, \iota )$ coincide (as $n'$-simplices of the simplicial set $\mathscr {F}(D)$). Since both $\alpha _{D}$ and $\alpha '_{D}$ are natural in $D$, we may assume without loss of generality that $D = C_{n'}$ and that $f$ is the identity morphism. In this case, the identities $T_{\operatorname{\mathcal{E}}}( \alpha ) = \tau = T_{\operatorname{\mathcal{E}}}( \alpha ' )$ give $\alpha _{D}(f,\iota ) = \tau _{n'} = \alpha '_{D}(f,\iota )$. $\square$