Why is this functor being described as a family of objects?

Question

Picture below from Mac Lane's Category Theory:

Why is this description of objects of the category $C^X$ valid?

By definition of the functor category it should be Objects : Functors $F,G,H, ... :X \rightarrow C $ and Arrows: $\tau, \tau' ...: F \rightarrow G$ as natural transformations.

Where the functors $F=(\text{F}_{\text{ob}}, \text{F$_{\text{arr}}$})$ have $F_{ob} : x \rightarrow F(x) \in C$ and $F(1_x) = 1_{F(x)}$ (since the only arrows in $x$ are the identity arrows).

So it would seem that the objects of $C^X$ are $\{F(x) \in C : x \in X, F \in \text{Funct}(X,C)\}$

But why is the below image description of the objects of $C^X$ a valid description?

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Answer 1

An object of $C^X$ is indeed a functor $X\rightarrow C$. However, think about what a functor from $X$ to $C$ is. Since $X$ is the discrete category, a functor $F:X\rightarrow C$ is entirely determined by what objects of $C$ it sends the objects of $F$ to: if $F_0, F_1$ are functors from $X$ to $C$ such that for all $x\in Ob(X)$ we have $F_0(x)=F_1(x)$, then in fact $F_0=F_1$. (Note that this is extremely false in general.)

So a functor $F: X\rightarrow C$ is essentially just the indexed family $\{a_x: x\in X, a_x=F(x)\}$: each completely determines the other. (This may get a bit more intuitive when you remember that set-theoretically, we generally think of a function as the corresponding set of ordered pairs - maybe we also explicitly add the codomain to this data, but in our case the codomain of the functor is always $C$, so we don't need to.) Moreover, any indexed family $\{a_x: x\in X\}$ corresponds to a functor $X\rightarrow C$. So it is in fact perfectly fine (if slightly notation-abusive) to say that the objects of $C^X$ are exactly the indexed families $\{a_x: x\in X\}$, since each such indexed set determines a unique functor and each functor corresponds to a unique such indexed family.

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Note that the indexing here is important: the range of $F$ isn't enough to determine $F$, we need to keep track of which object of $X$ go to a given object of $C$.

So you really want to think of an indexed family as a family of ordered pairs.

Answer 2

If $X$ is a set, then an $X$-indexed family of objects of $C$ is by definition a function $a$ that associates to any element $x$ of $X$ an object $a_x$ of $C$. It is not simply a set of objects of $C$ with the same cardinality as $X$.

Answer 3

A set $X$ can be turned into a (discrete) category by taking the elements of $X$ to be the objects and by taking only identities as morphisms.

With this identification, functors $X \to \mathcal{C}$ correspond exactly with $X$-indexed families of objects of $\mathcal{C}$; the action on morphisms is determined by the fact that the only morphisms of $X$ are identities.

Natural transformations between functors $X \to \mathcal{C}$ then correspond with $X$-indexed families of morphisms: the naturality condition is satisfied trivially again since the only morphisms in $X$ are identities.