Warning 7.3.6.2. In classical category theory, some authors take the universal property of Proposition 7.3.6.1 as the definition of a Kan extension. Beware that this is a slightly different notion in general: it is possible for a natural transformation $\beta : F_0 \rightarrow F \circ \delta $ to satisfy the universal property of Proposition 7.3.6.1 without exhibiting $F$ as a left Kan extension of $F_0$ along $\delta $ (in which case $F_0$ cannot admit any other left Kan extension along $\delta $; see Corollary 7.3.6.5).

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