??so let x be a tag theory and t a commutatively monoided topos ... ??then consider... ???

??"commutatively monoided topos equipped with x-model in its action tag environment" ... ???...

???given a commutatively monoided topos t, consider its action tag environment ...

???given a topos t' over t, where t is a particular commutatively monoided topos ... ???....

t' -> t -> _comm monoid_ ...

??we can thus still obtain a tag environment ... ??? t' becomes commutatively monoided ... and so forth .... ???then what about some sort of adjoint to this process??? .... and so forth .... ????.....


???something about ...???fixed "external" commutative monoid m ... process of getting tag environment from arbitrary topos x by taking m-actions in it ... ???or something?? ... and so forth ... ??.... ??something about adjoint to this process ... ???.....

??something about ... ???"classifying topos for t-model over m" for t a tag theory ... ??? and so forth ....

??so what about something about "strong vs lax" here (... ) ??? ....