I currently work on non-category theoretical research but use some tools from category theory. I am not well-versed in category theory and also don't want to expand the preliminaries of my work too much so I am looking for a reference to cite:
In this wikipedia article https://en.wikipedia.org/wiki/Universal_property#Relation_to_adjoint_functors I found the following result: Let $\mathcal{C}$ and $\mathcal{D}$ be categories and $F:\mathcal{C} \to \mathcal{D}$ be a functor.
Suppose that for any object $X$ of $\mathcal{D}$ there is a universal morphism $(G(X),u_X)$. Now let $X,Y$ be objects of $\mathcal{D}$ and $h:X\to Y$ a morphism. Then $f:= u_Y \circ h: X \to F(G(Y))$ and by definition of universal morphisms there is a unique morphism $G(h) : G(X) \to G(Y)$ such that $f= F(G(h)) \circ u_X$.
(Result) Then $G$ is a left-adjoint functor to $F$.
It would help me very much to get a reference to a book in which a proof of that result (Result) is given, so that I can a) read the proof in order to learn it and b) provide the reference to the readers of my work.
I have found this question Equivalence of the definition of Adjoint Functors via Universal Morphisms and Unit-Counit which seems to come close to mine but sadly there is no reference given.
Thank you and best regards
This is Theorem 2(ii) in Chapter IV of Saunders Mac Lane's 1971 book Categories for the Working Mathematician. It appears as exercise F in chapter 3 of Peter Freyd's 1965 book "An Introduction ot the Theory of Functors". Presumably it first appears in Freyd's unpublished thesis from 1960 since it is integral in the proof of his celebrated adjoint functor theorems.