CatGloss

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} $ .

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