The following are taken from "Arrows, Structures and Functors the categorical imperative" by Arbib and Manes.
$\color{Green}{Background:}$
$\textbf{Definition for “free over an object with respect to a functor":}$ Let $G:\textbf{A}\to \textbf{B}$ be any functor, and $B$ an object of $\textbf{B}.$ We say the pair $(A,\eta),$ where $A$ is an object of $\textbf{A}$ and $\eta:B\to GA$ is a morphism of $\textbf{B},$ is $\textbf{free over}$ $B$ $\textbf{with respect to}$ $G$ just in case $\eta;B\to GA$ has the couniversal property that given any morphism $f:B\to GA'$ with $A'$ any object of $\textbf{A,}$ there exists a unique $\textbf{A-}$morphism $\psi:A\to A';$ such that
Diagram 1
We refer to $\eta$ as the $\textbf{inclusion of generators;}$ and call the unique $\psi$ satisfying the above diagram the $\textbf{A-morphic extension of } f$ (with respect to $G$).
$\vdots$
We continue this section by showing that if every $B$ has a free $(A,\eta)$ with respect to the functor $G:\textbf{A}\to\textbf{B},$ then we can introduce a functor $F:\textbf{B}\to \textbf{A}$ for which $FB$ is free over $B.$ Such a functor $F$ is called a $\textbf{left adjoint}$ of $G.$ Dually, if every $B$ has a cofree object, then there is a cofree object-constructing functor called the $\textbf{right adjoint}$ of $G.$ Thus we say that $G$ $\textbf{has a left adjoint}$ just in case every object $B$ has a pair $(A,\eta)$ free over $B;$ and that $F$ $\textbf{has a right adjoint}$ if every $A$ has a pair $(B,\eta)$ cofree over $A.$
$\textbf{Theorem:}$ Let $G:\textbf{A}\to\textbf{B}$ be a functor with the property that to every $B$ in $\textbf{B}$ there corresponds a free object, call it $(FB,\eta B)$ ($\eta$ is the "inclusion of generators" map) [so that $\eta B:B\to GFB$]. Given any $f:B\to B'$ in $\textbf{B},$ define a morphism $Ff:FB\to FB'$ by the diagram
Then the collection of maps $F:\textbf{B}\to \textbf{A}$ so defined is a functor (clearly the object map is unique up to isomorphism, and fixes the morphism maps uniquely) call the $\textbf{left adjoint of}$ $G.$
$\color{Red}{Questions:}$
If $G:\textbf{A}\to\textbf{B},$ is a forgetful functor where $\textbf{A}$ is some category and $\textbf{B}=\textbf{Set},$ and I want to show that it has left adjoint or show the existence of left adjoint. (Sorry, I don't know what the proper phrasing is) Do I just need to show the existence of free objects over $\text{B}$ with respect to functor $G$ as in definition above or do I need to construct the free functor $F$ and show that it satisfies its universal mapping properties like in theorem 1 above. I actually don't know under what circumstances where only the above definition is needed or when I need to make use of the theorem above. Also there are dual versions to the above theorem and definition. My questions is the same thing for the case of right adjoint and cofree objects.
Thank you in advance.
This question is exactly answered by the theorem you have just cited: you only need to show that free objects exist and then the theorem tells you that you get the action of the left adjoint on morphisms for free. The situation is exactly the same for the dual question for cofree objects and right adjoints, by duality.
And in fact people implicitly use this theorem all the time; you will almost never see someone bother to define a left or a right adjoint on morphisms, and this is why.