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Let $F: C \to C'$ be a functor and let $A \in C'$.
The category $C_A$ is given by $\text{Ob}(C_A) = \{(X, s); X \in C, s : F(X) \to A \}$ and $\text{Hom}_{C_A}((X,s), (Y, t)) = \{ f \in \text{Hom}_C(X,Y); s = t\circ F(f)\}$.
But then after explaining Yoneda lemma and the convention $X(Y) ...
In mathematics, particularly category theory, a representable functor is a functor of a special form from an arbitrary category into the category of sets. Such functors give representations of an abstract category in terms of known structures (i.e. sets and functions) allowing one to utilize, as much as possible, knowledge about the category of sets in other settings.
From another point of view, representable functors for a category C are the functors given with C. Their theory is a vast generalisation of upper sets in posets, and of Cayley's theorem in group theory.
== Definition ==
Let C be a...
