Left Kan Extension along identity?

Solution 1:

First of all, it is $(1 \downarrow \star)$, not $(\phi \downarrow \star).$

Now, this should look very suspicious because the indexing category $(1 \downarrow c)$ has a terminal object $1_c$, so taking colimit over it should not collapse anyone anywhere. Another way to see it is that the adjunction $\operatorname{Nat}(\operatorname{Lan}_KF, G) = \operatorname{Nat}(F, G \circ K)$ for $K = 1$ means that we should have $\operatorname{Lan}_KF \simeq F$.

In what you wrote everything looks fine until the moment of taking the colimit. The way you tried to take it is not quite correct, as you should, informally, factor by some relation a union of copies of $A$ indexed by the indexing diagram, not factor the image of the diagram.

So the moral is that the colimit over $\bullet \to \bullet$ has to be the image of the terminal object. You can take your favorite approach to colimit computation and accurately ensure that it agrees with this statement.