Is the comma category $y \downarrow X$ small?

Question

Let $\mathcal{C}$ be a small category and $X \in \hat{\mathcal{C}}$ a presheaf. Is the comma category $y \downarrow X$ small?

Answer 1

I'm detailing Zhen Lin's comment.

Yoneda's lemma states (notably) that for every object $c$ of $\mathcal C$, there is a bijection $\hom(y(c),X) \simeq X(c)$.

But the object of $(y\downarrow X)$ are precisely the arrows $y(c) \to X$ for all object $c$ of $C$. Then the class of objects of $(y\downarrow X)$ are in bijection with $\coprod_{c \in \operatorname{Ob}\mathcal C} X(c)$ : each $X(c)$ is a small set, and $\operatorname{Ob}\mathcal C$ is also a small set, so is the disjoint sum.

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Following magma's suggestion, let's show that $(y \downarrow X)$ is locally small : the arrows in $(y\downarrow X)$ between the objects $y(c) \to X$ and $y(c') \to X$ are those arrow $f\colon c \to c'$ in $\mathcal C$ such that commutes $$ \begin{matrix} y(c) & \overset{y(f)} \longrightarrow & y(c') \\ \ \ \ \ \ \ \ \ \ \searrow & & \!\!\!\!\!\!\!\!\!\!\!\!\swarrow \\ & X. & \end{matrix} $$ So the $\hom(y(c) \to X,y(c')\to X) \subseteq \hom(c,c')$, and a subset of a small set is itself small.