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This is a follow up to the following two MO questions: q1,q2

What I'm interesting in understanding is the universal property (if any) of the morphism $Sum_n:Fam_n(\mathcal{C})\to \mathcal{C}$ considered by Freed-Hopkins-Lurie-Teleman in Topological Quantum Field Theories from Compact Lie Groups. Namely, the existence and the more or less explicit form of such a monoidal functor is in itself such an amazing fact that one one could easily forget that if just a monoidal functor $Fam_n(\mathcal{C})\to \mathcal{C}$ is what one seeks, then this problem would be trivially solved by the trivial (or zero) functor. This would be clearly a trivial solution and one could say that what is remarkable in FHLT cosnstruction is that it is not trivial, but this would be quite a weak statement. Rather I suspect (and the question is precisely "How much are these suspects correct and which are references for them?") that $Sum_n$ secretely satisfies an adjointness property. More precisely, one has the forgetful functor $Fam_n(\mathcal{C})\to Fam_n$ and so a fully extended tqft with target $Fam_n(\mathcal{C})$ determines a "classical background" $X$, as the image of the point via the composition $Bord_n \to Fam_n(\mathcal{C})\to Fam_n$ (here I'm omitting eventual framings and orientations fore ease of writing/reading). A lifting from $Bord_n \to Fam_n$ to $Bord_n \to Fam_n(\mathcal{C})$ is the datum of a $\mathcal{C}$-local system on $X$ )or at least, it should be, if I'm not mistaken here), and $\mathcal{C}$ itself can be identified with $\mathcal{C}$-local system on the point.

So from this point of view the $Sum_n$ construction seems to be a push-forward to the point of a $\mathcal{C}$-local system on $X$. This suggests to consider the more general situation of a morphism $f:X\to Y$ of classical backgrounds, and to consider the pull-back of $\mathcal{C}$-local systems (this should be the easy direction) and wondering whether this has an adjoint. Then $Sum_n$ in FHLT should map the $\mathcal{C}$-local system $A:X\to \mathcal{C}$ of the pushforward of $A$ along the terminal morphism $X\to *$, and all the other values of $Sum_n$ should be uniquely determined by the cobordism hypothesis.

Is this correct? References?

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