Twisted Comma Category

The twisted comma category of $\mathrm{Tw}(F,G)$ of two Functors $F:\mathcal{A}\to\mathcal{C}$, $G:\mathcal{B}\to\mathcal{C}$ is the category where

• Objects are triples $X:\mathcal{A},Y:\mathcal{B},f:{{F(X)}\to{G(Y)}}$

• Morphisms are squares of the following form

Properties

• Twisted comma categories are Discrete Cocartesian Fibrations when viewed as Displayed Categories over $\mathcal{A}^{op}\times\mathcal{B}$
• The Twisted Arrow Category is the twisted comma category $\mathrm{Tw}(mathrm{Id}{ifx..lse\par i},mathrm{Id}{ifx..lse\par i})$.