Given a category $C$ and two diagrams $D:I→C$ and $E:J→C$, suppose the image of $I$ and $J$ are not overlapped and we know that $D$ has limit, $E$ has colimit, i.e. $lim D$ and $colim E$. Now I can take the product of the two apex of $lim D$ an $colim E$, i.e. $lim D \times colim E$ (suppose the product exists).
The question is:
Is there any theory about this kind of construction $lim D \times colim E$? (It mix limit and colimt.)
Is there any relationship between $lim D \times colim E$ and some universal construction on the diagram $D+E$? The diagram $D+E$ means the obvious functor indexed by $I+J$ (the coproduct of $I$ and $J$, which has a copy of $I$ and a copy of $J$ and no arrows between them).
Background:
This question inspired by the previous quesiton Does the limit(colimit) have transitivity?. In that quesiton, following Maxime Ramzi's answer, we have a conclusion. That is $lim (D+E)$ exists if and only if $lim D × lim E$ exists, in which case there is a canonical isomorphism between the two. But here, we take $colim E$ instead of $lim E$.
Thanks.