?? so let's try getting somewhat of a feel for left adjoint of cocontinuous extension of cartesian product in presheaf topos .... ???? ....

?? first, just plain cocontinuous extension of cartesian product ....

?? graph ....

?? "bi-graph" ...

?? "realize bi-graph as graph by using vertex for vertex-vertex, vertex pair for edge-vertex and vertex-edge, and ... edge^2 for edge-edge" .... ???? ...

?? right adjoint here as .... ?? ....


universal (?? generic ?? ...) bicocontinuous followed by cocontinuous = bicocontinuous .... ???? ... ??? adjoints to these ??? ....

aXb -> a#b -> c

?? level slip here ... "cartesian product" ... as functor with domain a cartesian product ... ??? ....

_graph_ X _graph_ -> _bigraph_ -> _graph_


?? "restrict bigraph along :

walking graph category -> walking bigraph category

to get graph" .... ??? ....

?? is this coming out the same as what we get from other description above ?? .... ??? what's going on here ??? .... ??? vaguely reminds me of ... ?? some stuff people say about simplicial objects ... ??? ....