Kerodon

Warning 5.4.7.8. For every pair of simplicial sets $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$, we let $\operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$ denote the simplicial set introduced in Construction 1.5.3.1. When working with $(\infty ,2)$-categories, this notation is potentially confusing. By construction, vertices of the simplicial set $\operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$ can be identified with morphisms of simplicial sets $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$. If $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ are $(\infty ,2)$-categories, then such morphisms need not carry thin $2$-simplices of $\operatorname{\mathcal{C}}$ to thin $2$-simplices of $\operatorname{\mathcal{D}}$, and therefore need not correspond to functors from $\operatorname{\mathcal{C}}$ to $\operatorname{\mathcal{D}}$ in the sense of Definition 5.4.7.1. We will return to this point in §.