| To do this, we need a definition. Given categories and functors $ \xymatrix{ &\mathscr{B} \ar[d]^Q \ \mathscr{A} \ar[r]_P & \mathscr{C}, } $ the comma category $ P {\mathbin{\Rightarrow} Q} $ (often written as $ (P \mathbin{\downarrow} Q) $ ) is the category defined as follows: Given $ \mathscr{A} $ , $ \mathscr{B} $ , $ \mathscr{C} $ , $ P $ and $ Q $ as above, there are canonical functors and a canonical natural transformation as shown: $ \xymatrix{ P {\mathbin{\Rightarrow} Q} \ar[r] \ar[d] \ar@{}[dr] | - {\rotatebox{45 } { $ \Rightarrow $ }} & \mathscr{B} \ar[d]^Q \ \mathscr{A} \ar[r]_P & \mathscr{C} } $ In a suitable 2 - categorical sense, $ P {\mathbin{\Rightarrow} Q} $ is universal with this property. Example: Let $ \mathscr{A} $ be a category and $ A \in \mathscr{A} $ . |