Kerodon

Remark 8.1.3.17. Let $\operatorname{\mathcal{C}}$ be a category which admits pushouts. The construction of the $2$-category $\operatorname{Cospan}(\operatorname{\mathcal{C}})$ of Example 2.2.2.1 involves some auxiliary choices: if $X \rightarrow B \leftarrow Y$ and $Y \rightarrow C \leftarrow Z$ are cospans in $\operatorname{\mathcal{C}}$, then their composition (as $1$-morphisms of $\operatorname{Cospan}(\operatorname{\mathcal{C}})$) is given by $X \rightarrow (B \amalg _{Y} C) \leftarrow Z$, where the pushout $B \amalg _{Y} C$ is only well-defined up to (canonical) isomorphism. Corollary 8.1.3.15 supplies a description of the Duskin nerve $\operatorname{N}_{\bullet }^{\operatorname{D}}( \operatorname{Cospan}(\operatorname{\mathcal{C}}) )$ which does not depend on these choices. This shows, in particular, that the $2$-category $\operatorname{Cospan}(\operatorname{\mathcal{C}})$ is well-defined up to (non-strict) isomorphism; see Example 2.2.6.13.