Remark 4.3.2.18. Let $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{E}}$ be a functor between categories. According to Remark 4.3.1.11, the slice category $\operatorname{\mathcal{E}}_{/G}$ can be identified with the iterated fiber product
\[ (\operatorname{Fun}( [0], \operatorname{\mathcal{E}}) \times _{ \operatorname{Fun}( \{ 0\} \times \operatorname{\mathcal{D}}, \operatorname{\mathcal{E}}) } \operatorname{Fun}( [1] \times \operatorname{\mathcal{D}}, \operatorname{\mathcal{E}}) ) \times _{ \operatorname{Fun}( \{ 1\} \times \operatorname{\mathcal{D}}, \operatorname{\mathcal{E}}) } \{ G \} . \]
Using Example 4.3.2.15, we can identify the left factor with the functor category $\operatorname{Fun}( \operatorname{\mathcal{D}}^{\triangleleft }, \operatorname{\mathcal{E}})$. We therefore obtain a pullback diagram of categories
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{E}}_{/G} \ar [r] \ar [d] & \operatorname{Fun}( \operatorname{\mathcal{D}}^{\triangleleft }, \operatorname{\mathcal{E}}) \ar [d] \\ \{ G\} \ar [r] & \operatorname{Fun}( \operatorname{\mathcal{D}}, \operatorname{\mathcal{E}}), } \]
which recovers Example 4.3.2.11 at the level of objects.
Similarly, if $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{E}}$ is a functor of categories, then the coslice category $\operatorname{\mathcal{E}}_{F/}$ fits into a pullback square
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{E}}_{F/} \ar [r] \ar [d] & \operatorname{Fun}( \operatorname{\mathcal{C}}^{\triangleright }, \operatorname{\mathcal{E}}) \ar [d] \\ \{ F\} \ar [r] & \operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{E}}). } \]