Remark 5.2.3.3 (0244)

Remark 5.2.3.3. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{E}}$ and $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{E}}$ be morphisms of simplicial sets. Then the inclusion maps $\operatorname{\mathcal{C}}\hookrightarrow \operatorname{\mathcal{C}}\star \operatorname{\mathcal{D}}\hookleftarrow \operatorname{\mathcal{D}}$ lift uniquely to monomorphisms

\[ \iota _{\operatorname{\mathcal{C}}}: \operatorname{\mathcal{C}}\hookrightarrow \operatorname{\mathcal{C}}\star _{\operatorname{\mathcal{E}}} \operatorname{\mathcal{D}}\quad \quad \iota _{\operatorname{\mathcal{D}}}: \operatorname{\mathcal{D}}\hookrightarrow \operatorname{\mathcal{C}}\star _{\operatorname{\mathcal{E}}} \operatorname{\mathcal{D}}, \]

which fit into a commutative diagram

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{C}}\ar [r]^-{\iota _{\operatorname{\mathcal{C}}}} \ar [d] & \operatorname{\mathcal{C}}\star _{\operatorname{\mathcal{E}}} \operatorname{\mathcal{D}}\ar [d] & \operatorname{\mathcal{D}}\ar [l]_-{\iota _{\operatorname{\mathcal{D}}}} \ar [d] \\ \{ 0\} \ar [r] & \Delta ^1 & \{ 1\} \ar [l] } \]

in which both squares are pullbacks. In the future, we will often abuse notation by identifying $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ with their images under the monomorphisms $\iota _{\operatorname{\mathcal{C}}}$ and $\iota _{\operatorname{\mathcal{D}}}$, respectively (which are full simplicial subsets of the relative join $\operatorname{\mathcal{C}}\star _{\operatorname{\mathcal{E}}} \operatorname{\mathcal{D}}$).