• For every functor we have the implications

$F$ is very rich $\To$ $F$ is rich (Hilton) $\To$ $F$ is opaque $\To$ $F$ is consistent (Frei)

and each converse implication is false in general.

• An opaque functor is consistent and co-consistent (i.e. $F^\op$ is consistent).
• If $L\dashv U$ is an adjunction, $U$ is opaque if and only if $L$ is opaque.
• If $L\dashv U$ is an adjunction, $U$ is very rich if and only if it is rich, if and only if it is opaque, if and only if it is consistent.