Let

$F:A^{\mbox{op}} \to \mbox{Set}$

and define

$G_a:A\times A^{\mbox{op}} \to \mbox{Set}$

$G_a(b,c) = \mbox{hom}(a,b) \times F(c)$.

I think the coend of $G_a$,

$\int^AG_a$,

ought to be $F(a)$--it's certainly true when A is discrete, since then hom is a delta function. But my colimit-fu isn't good enough to actually compute the thing and verify it's true. Can someone walk me through the computation, please?