The category of elements $\textstyle{\int}!{el}$ of a contravariant functor $F \colon \mathsf{C}^\mathrm{op} \to \textup{\textsf{cat}}$ has
- as objects, pairs $(c,x)$ where $c \in \mathsf{C}$ and $x \in Fc$, and
- a morphism $(c,x) \to (c’,x’)$ is a morphism $f \colon c \to c’$ in $\mathsf{C}$ so that $Ff(x’) = x$.
The category of elements has an evident forgetful functor $\Pi \colon \textstyle{\int}!{el} \to \mathsf{C}$.
\(\xymatrix{ \mathrm{If}\ Ff(x')=x,\ \mathrm{then} & (c,x) \ar[r]^f \ar@{}[dr]\mid{\rotatebox{-90}{$\mapsto$}}^(.6){\Pi} & (c',x') & \in & \textstyle{\int}\!{el} \ar[d]^\Pi \\ & c \ar[r]_f & c' & \in & \mathsf{C}}\)