?? "allegory" .... ?? with factorization system of relation into co-monic followed by monic .... ??? ... ?? getting allegory (?? ...) from lawvere theory (?? ...) .... ???? .... ?? niceness (?? ...) of such process ?? ...

no wait ... confusion about "joint monic" vs "placewise monic" ... ?? factorization of relation into co-map followed by map .... ???? .... ....

relation as span ... ?? ...

?? matrix treated sesquivariantly : bistable bimonoid : matrix treated bicovariantly : "double stable monoid" .... ??? .... ???? .... just stupid guess ... ?? ...

?? "quantifier sequence" .... ??? arising when both adjoints are available ....
?? when only one is available, then maybe much simpler .... ???? ...