Let $U$ be an object of $\mathcal{C}$. We write $\underline{U}$ for the functor $\mathop{\mathrm{Mor}}\nolimits _\mathcal {C}(U, -): \mathcal{C} \to \textit{Sets}$. This defines a fully faithful embedding of $\mathcal C^{opp}$ into the category of functors $\mathcal{C} \to \textit{Sets}$. Hence, if $f : U \to V$ is a morphism, we are justified in denoting still by $f$ the induced morphism $\underline{V} \to \underline{U}$, and vice-versa.
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