Remark 1.7.3. If \(I\) and \(C\) are 1-categories, then so is \(\Fun (I,C)\), and hence all animae occurring in Definition 1.7.1 are sets. The condition on a cone \((X,\epsilon )\) then says precisely that for every object \(Y\) the map of sets \[ \Hom _C(Y,X) \to \Nat (\const _Y,F), \qquad u \mapsto \epsilon \circ \const _u, \] is a bijection, that is, that every cone on \(F\) with tip \(Y\) factors uniquely through \((X,\epsilon )\). So Definition 1.7.1 recovers the classical notion of a limit, and dually of a colimit, in a 1-category.
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