Theorem 3.3. Assuming that:

$G : D \to C$ is a functor
for $A \in ob C$ , let $(A ↓ G)$ be the category whose objects are pairs $(B, f)$ where $B \in ob D$ and $f : A \to G B$ , and whose morphisms $(B, f) \to (B^{'}, f^{'})$ are morphisms $g : B \to B^{'}$ making
commute.

Then specifying a left adjoint for

G

is equivalent to specifying an initial object of

(A ↓ G)

for each

A