Are cofibrations accessible?

A cofibrantly generated $(L,R)$ does not need to have $L$ accessible, see Example 3.5 in my paper "On combinatorial model categories." Also, $L$ accessible does not imply that $(L,R)$ is cofibrantly generated, even accessible. Take regular monos in Boolean algebras. This $L$ is accessible but $(L,R)$ cannot be accessible because regular injectives are complete Boolean algebras which are not accessible.