pasting along an adjunction

[Andrew Salch <asalch@math.jhu.edu>]

Let C,D be categories, let F be a functor from C to D, and let G be right
adjoint to F. In a recent paper of Connes and Consani, they consider the
following "pasting along an adjunction": let E be a category whose object
class is the union of the object class of C and the object class of D; and
given objects X,Y of E, let the hom-set hom_E(X,Y) be defined as follows:

-if X,Y are both in the object class of C, then hom_E(X,Y) = hom_C(X,Y).

-if X,Y are both in the object class of D, then hom_E(X,Y) = hom_D(X,Y).

-if X is in the object class of C and Y is in the object class of D, then
hom_E(X,Y) = hom_C(X,GY) = hom_D(FX,Y).

-if X is in the object class of D and Y is in the object class of C, then
hom_E(X,Y) is empty.

Composition is defined in a straightforward way. When C,D are closed
symmetric monoidal categories, then E has a natural closed symmetric
monoidal structure as well. Connes and Consani use this categorical
pasting to construct schemes over F_1, "the field with one element," and I
have worked out some variations and applications of this categorical
pasting which produce other useful objects (e.g. algebraic F_1-stacks and
derived F_1-stacks, which have some useful number-theoretic as well as
homotopy-theoretic properties).

I would like to know if this "pasting along an adjunction" is a special
case of some more general construction already known to category theory,
and if basic properties of pasting along an adjunction have already been
worked out and written down somewhere.

Thanks,
Andrew S.