Construction 5.48.
Let \(\Uu\) and \(\Zz\) be categories with finite limits, and let \(F\colon \Uu \to \Zz\) preserve terminal objects. We define the category \(\Xx\) as the comma category of \(F\):
\[\Xx \quad := \quad \Uu \times_{\Zz,\,F} \Ar(\Zz),\]
where the functor \(\Ar(\Zz)\to\Zz\) is evaluation at the target. In other words, the objects of \(\Xx\) are triples \((U,Z,\varphi)\) where \(U \in \Uu\), \(Z \in \Zz\) and \(\varphi\colon Z \to F(U)\).
Note that the two forgetful functors \(j^*\colon \Xx \to \Uu\) and \(i^*\colon \Xx \to \Zz\) admit canonical right adjoints
\[j_*\colon \Uu \hookrightarrow \Xx, U \mapsto (U,F(U),\id_{F(U)}), \qquadtext{ and } i_*\colon \Zz \hookrightarrow \Xx, Z \mapsto (*,Z,Z\to F(*)\iso *).\]