Regular Span

A span of categories $A\leftarrow E\rightarrow B$ is a regular span if it meets the first 2 conditions of a 2-Sided Fibration. Namely:

• For every $u:A(a,b)$, $x:B$, and $e:E_{b,x}$, there exists a $B$ vertical Cartesian lift $(u^{*}(e):E_{a,x},f:E_{u,\mathrm{id}}(u^{*}(e),e))$
• For every $a:A$, $x,y:B$, and $e:E_{a,x}$, there exists an $A$ vertical Cocartesian lift $(u_{!}(e):E_{a,y},f:E_{\mathrm{id},u}(e,u_{!}(e)))$.