Theorem 4.12
(General Adjoint Functor Theorem)
.
Assuming that:
D
is
complete
and
locally small
Then
G
:
D
→
C
has a
left adjoint
if and only if
G
preserves
small
limits
and satisfies the
solution-set condition
: for every
A
∈
ob
C
, there’s a set
{
(
B
i
,
f
i
)
|
i
∈
I
}
of objects of
(
A
↓
G
)
which is collectively
weakly
initial
.