Example 4.3.3.7 (016S)

Example 4.3.3.7. For every category $\operatorname{\mathcal{C}}$, let $h_{\operatorname{\mathcal{C}}}: \operatorname{Lin}^{\operatorname{op}} \rightarrow \operatorname{Set}$ denote the functor represented by $\operatorname{\mathcal{C}}$, given by the formula

\[ h_{\operatorname{\mathcal{C}}}(J) = \{ \textnormal{Functors from $J$ to $\operatorname{\mathcal{C}}$} \} . \]

For every pair of categories $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ and every finite linearly ordered set $J$, we have a canonical bijection

\begin{eqnarray*} (h_{\operatorname{\mathcal{C}}} \star h_{\operatorname{\mathcal{D}}})(J) & = & \coprod _{ I \sqsubseteq J} h_{\operatorname{\mathcal{C}}}( I) \times h_{\operatorname{\mathcal{D}}}(J \setminus I) \\ & = & \coprod _{I \sqsubseteq J} \{ \textnormal{Functors $I \rightarrow \operatorname{\mathcal{C}}$} \} \times \{ \textnormal{Functors $(J \setminus I) \rightarrow \operatorname{\mathcal{D}}$} \} \\ & \simeq & \{ \textnormal{Functors $J \rightarrow \operatorname{\mathcal{C}}\star \operatorname{\mathcal{D}}$} \} \\ & = & h_{\operatorname{\mathcal{C}}\star \operatorname{\mathcal{D}}}(J). \end{eqnarray*}

These bijections depend functorially on $J$, and therefore determine an isomorphism $h_{\operatorname{\mathcal{C}}} \star h_{\operatorname{\mathcal{D}}} \simeq h_{\operatorname{\mathcal{C}}\star \operatorname{\mathcal{D}}}$ in the category $\operatorname{Fun}( \operatorname{Lin}^{\operatorname{op}}, \operatorname{Set})$; here $\operatorname{\mathcal{C}}\star \operatorname{\mathcal{D}}$ denotes the join of the categories $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$, in the sense of Definition 4.3.2.1.