  A novel functorial relationship in perturbative quantum field theory is
pointed out that associates Feynman diagrams (FD) having no external line in
one theory ${\bf Th}_1$ with singlet operators in another one ${\bf Th}_2$
having an additional $U({\cal N})$ symmetry and is illustrated by the case
where ${\bf Th}_1$ and ${\bf Th}_2$ are respectively the rank $r-1$ and the
rank $r$ complex tensor model. The values of FD in ${\bf Th}_1$ agree with the
large ${\cal N}$ limit of the Gaussian average of those operators in ${\bf
Th}_2$. The recursive shift in rank by this FD functor converts numbers into
vectors, then into matrices, and then into rank $3$ tensors ${\ldots}$ This FD
functor can straightforwardly act on the $d$ dimensional tensorial quantum
field theory counterparts as well. In the case of rank 2-rank 3 correspondence,
it can be combined with the geometrical pictures of the dual of the original
FD, namely, equilateral triangulations (Grothendieck's dessins d'enfant) to
form a triality which may be regarded as a bulk-boundary correspondence.
