Corollary 5.2.7.4. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a morphism of simplicial sets. The following conditions are equivalent:
- $(1)$
There exists a pullback diagram of simplicial sets
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{E}}\ar [r] \ar [d]^{U} & \operatorname{N}_{\bullet }( \operatorname{\mathcal{D}}) \ar [d]^{ \operatorname{N}_{\bullet }(V) } \\ \operatorname{\mathcal{C}}\ar [r] & \operatorname{N}_{\bullet }( \mathrm{h} \mathit{\operatorname{\mathcal{C}}} ), } \]where $V: \operatorname{\mathcal{D}}\rightarrow \mathrm{h} \mathit{\operatorname{\mathcal{C}}}$ is a left covering functor (in the sense of Definition 4.2.3.1).
- $(2)$
For every category $\operatorname{\mathcal{C}}'$ and every morphism of simplicial sets $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}') \rightarrow \operatorname{\mathcal{C}}$, the fiber product $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}') \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}$ is isomorphic to the nerve of a category $\operatorname{\mathcal{E}}'$ and the projection $\operatorname{\mathcal{E}}' \rightarrow \operatorname{\mathcal{C}}'$ is left covering functor (in the sense of Definition 4.2.3.1).
- $(3)$
For every $n$-simplex $\sigma : \Delta ^ n \rightarrow \operatorname{\mathcal{C}}$, the fiber product $\Delta ^ n \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}$ is isomorphic to the nerve of a category $\operatorname{\mathcal{E}}'$ and the projection $\operatorname{\mathcal{E}}' \rightarrow [n]$ is a left covering functor (in the sense of Definition 4.2.3.1).
- $(4)$
The morphism $U$ is a left covering map of simplicial sets (in the sense of Definition 4.2.3.8).