Morphisms between objects are in bijection with natural transformations between the functors they represent.

Lemma 59.9.3 (Yoneda). Let $\mathcal{C}$ be a category, and $X, Y \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$. There is a natural bijection

$\begin{matrix} \mathop{\mathrm{Mor}}\nolimits _\mathcal {C}(X, Y) & \longrightarrow & \mathop{\mathrm{Mor}}\nolimits _{\textit{PSh}(\mathcal{C})} (h_ X, h_ Y) \\ \psi & \longmapsto & h_\psi = \psi \circ - : h_ X \to h_ Y. \end{matrix}$

Proof. See Categories, Lemma 4.3.5. $\square$

Comment #1281 by on

Suggested slogan: Morphisms between objects are in bijection with natural transformations between the functors the represent.

There are also:

• 2 comment(s) on Section 59.9: Presheaves

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).