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The Stacks project

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

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