Example 7.2.5.3. Let $\operatorname{\mathcal{C}}$ be an ordinary category, which we regard as an $\mathrm{h} \mathit{\operatorname{Kan}}$-enriched category in which each of the Kan complexes $\underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(X,Y)$ is equal to $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)$ (regarded as a constant simplicial set). In this case, condition $(\ast _ n)$ of Definition 7.2.5.1 is automatically satisfied for $n \geq 3$. Moreover, we can state conditions $(\ast _1)$ and $(\ast _2)$ more concretely as follows:
- $(\ast _1)$
For every pair of objects $X,Y \in \operatorname{\mathcal{C}}$, there exists an object $Z \in \operatorname{\mathcal{C}}$ equipped with morphisms $u: X \rightarrow Z$ and $v: Y \rightarrow Z$.
- $(\ast _2)$
For every pair of objects $X,Y \in \operatorname{\mathcal{C}}$ and every pair of morphisms $f_0, f_1: X \rightarrow Y$, there exists a morphism $v: Y \rightarrow Z$ satisfying $v \circ f_0 = v \circ f_1$.
It follows that $\operatorname{\mathcal{C}}$ is homotopy filtered (in the sense of Definition 7.2.5.1) if and only if is filtered (in the sense of Definition 7.2.4.1).