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I don't know of any definition involving Kan extensions, but (co)ends can be expressed as (co)limits weighted by a hom functor (see e.g. here), so that for $F \colon C^{op} \times C \to D$ the end $\int_c F(c,c)$ is $\lim^{\hom_{C}} F$, where hom is taken as a profunctor $C^{op} \times C \not\to 1$. Similarly for (pointwise) Kan extensions: $(\operatorname{Ran}_K G)(c) = \lim^{C(c,K-)} G$, where $C \overset{K}{\leftarrow} B \overset{G}{\to} D$, so that the weight is the 'representable' profunctor $C(1,K) \colon B \not\to C$. See e.g. Emily Riehl's notes referenced here.