I'm reading Tom Leinster's "Basic Category Theory" (self-study).
In the book p.$63$, the author mentions a corollary:
This corollary means that if for each $A∈\mathcal{A}$, we have a initial morphism from $A \to G$, then $F \dashv G$ (No need to declare these initial morphisms to be natural).
In his proof, he let $F(f) : F(A) \to F(A')$ be the unique map.
But why we can claim that the $F(f)$ is exactly the unique map?
AFAIK, if $\eta_A : A \to GF(A)$ is initial morphism, then for any candidate $\bar{q} : A \to G(B)$ (in this case, $\eta_{A'} \circ f : A \to GF(A')$), there is unique morphism $q:F(A) \to B$ (in this case, $q:F(A) \to F(A')$), such that $G(q) \circ \eta_A = \bar{q}$ (in this case, $G(q) \circ \eta_A = \eta_{A'} \circ f $).
There seems to be no guarantee that $q:F(A) \to F(A')$ must be $F(f)$?
I may miss some key thing, can anyone give me some hints
Very thanks.
After some thinking, I think I can answer my own question.
This corollary really says that:
Given a functor $G : \mathcal{B} → \mathcal{A}$, if for each $A∈\mathcal{A}$, we have a initial morphism from $A→G$, then we can construct a functor $F : \mathcal{A} → \mathcal{B}$, such that $F \dashv G$.
The method of constructing F is as follows:
For $\forall f : A \to A'$,
F map object $A$ to "$F(A)$",
F map object $A'$ to "$F(A')$",
F map morphism $f$ to "$F(f)$".
The quotation marks mean that they are just the name of an object/morphism in $\mathcal{B}$.
It also means that following statement is wrong:
Given two functor $F$ and $G$, if for each $A∈\mathcal{A}$, we have $\eta_A : A \to GF(A)$, such that $\langle F(A), \eta_A \rangle$ is initial morphism, then $F \dashv G$.
This because there may be multiple functors (e.g. $F_1$, $F_2$, ...) which satisfy universal property, but may only $F_1$ be the left adjoint of G, $F_2$ not.
In other words, this corollary involves "The existence of left adjoint" rather than "what is an adjoint functors".