Example 5.6.1.9. Let $\mathscr {I}$ denote the inclusion from the ordinary category $\operatorname{Cat}$ (regarded as a $2$-category having only identity $2$-morphisms) to the $2$-category $\mathbf{Cat}$, and let $\operatorname{Cat}_{\ast }^{\mathrm{lax}}$ denote the category of elements $\int _{\operatorname{Cat}} \mathscr {I}$. The category $\operatorname{Cat}_{\ast }^{\mathrm{lax}}$ can be described concretely as follows:
The objects of $\operatorname{Cat}_{\ast }^{\mathrm{lax}}$ are pairs $(\operatorname{\mathcal{C}}, X)$, where $\operatorname{\mathcal{C}}$ is a category and $X$ is an object of $\operatorname{\mathcal{C}}$.
A morphism from $(\operatorname{\mathcal{C}}, X)$ to $(\operatorname{\mathcal{D}}, Y)$ is a pair $(F, u)$, where $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ is a functor and $u: F(X) \rightarrow Y$ is a morphism in the category $\operatorname{\mathcal{D}}$.
If $(F,u): (\operatorname{\mathcal{C}}, X) \rightarrow (\operatorname{\mathcal{D}}, Y)$ and $(G,v): (\operatorname{\mathcal{D}}, Y) \rightarrow (\operatorname{\mathcal{E}}, Z)$ are morphisms in $\operatorname{Cat}_{\ast }^{\mathrm{lax}}$, then their composition is the pair $(G \circ F, w)$, where $w$ is the morphism of $\operatorname{\mathcal{E}}$ given by the composition
\[ (G \circ F)(X) \xrightarrow { G(u) } G(Y) \xrightarrow {v} Z. \]