Lemma 4.3.2.9 (0169)

Lemma 4.3.2.9. Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be categories, and let $\iota _{\operatorname{\mathcal{C}}}: \operatorname{\mathcal{C}}\hookrightarrow \operatorname{\mathcal{C}}\star \operatorname{\mathcal{D}}$ and $\iota _{\operatorname{\mathcal{D}}}: \operatorname{\mathcal{D}}\hookrightarrow \operatorname{\mathcal{C}}\star \operatorname{\mathcal{D}}$ denote the inclusion maps. Then:

$(1)$: The inclusion functor $\iota _{\operatorname{\mathcal{C}}}$ factors uniquely as a composition

\[ \operatorname{\mathcal{C}}\xrightarrow { \overline{\iota }_{\operatorname{\mathcal{C}}} } (\operatorname{\mathcal{C}}\star \operatorname{\mathcal{D}})_{/ \iota _{\operatorname{\mathcal{D}}} } \rightarrow \operatorname{\mathcal{C}}\star \operatorname{\mathcal{D}}. \]
$(2)$: The inclusion functor $\iota _{\operatorname{\mathcal{D}}}$ factors uniquely as a composition

\[ \operatorname{\mathcal{D}}\xrightarrow { \overline{\iota }_{\operatorname{\mathcal{D}}} } (\operatorname{\mathcal{C}}\star \operatorname{\mathcal{D}})_{ \iota _{\operatorname{\mathcal{C}}} / } \rightarrow \operatorname{\mathcal{C}}\star \operatorname{\mathcal{D}}. \]

Proof. Let $\pi _{\operatorname{\mathcal{C}}}: \operatorname{\mathcal{C}}\times \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$ and $\pi _{\operatorname{\mathcal{D}}}: \operatorname{\mathcal{C}}\times \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{D}}$ denote the projection maps. Using Remark 4.3.1.11, we see that both $(1)$ and $(2)$ are equivalent to the assertion that there is a unique natural transformation $u$ from $\iota _{\operatorname{\mathcal{C}}} \circ \pi _{\operatorname{\mathcal{C}}}$ to $\iota _{\operatorname{\mathcal{D}}} \circ \pi _{\operatorname{\mathcal{D}}}$ (as functors from the product category $\operatorname{\mathcal{C}}\times \operatorname{\mathcal{D}}$ to the join category $\operatorname{\mathcal{C}}\star \operatorname{\mathcal{D}}$). Concretely, this natural transformation carries each object $(C,D) \in \operatorname{\mathcal{C}}\times \operatorname{\mathcal{D}}$ to the unique element of $\operatorname{Hom}_{\operatorname{\mathcal{C}}\star \operatorname{\mathcal{D}}}(C, D)$. $\square$