?? doctrine of categories with x-shaped colimits ... x some class of diagram schemes ... for example doctrine of filteredly cocomplete categories ...

?? _set_^op as environment here??? ... ??? ... ??variance conventions / issues ... ??? .....

???generalized yoneda embedding ????? ....

??? .... as "free cocompletion, relative to keeping original x-shaped colimits" .... ????....

???? ..... ???object in theory (???"formula" ???) st ..... ??? homming out of it preserves x-shaped colimits .... ???something about "representable formula" ???? .... "formula representable by model" ????? ..... ????? ..... ????? ....

?????doctrine of categories with x-shaped limits, vs doctrine of categories with x-shaped limits and [colimits over which they distribute] ???? ..... ????? .... ??? "monoid algebra" ????? ....... ???? "operad" ..... ????? ..... ??? ....????? .....