Weighted (co)limit, perhaps incomplete specification of the natural bijection

Question

I'm trying to understand the notion of weighted colimit in as detail as possible. Here is a good explanation. However in the first displayed formula on page 100 starting with $\cal M$, the isomorphisms is too "general" for me. I would expect that the isomorphism works by pre-composition with an arrow $m\to $lim$F$ from the set on the l.h.s. and the limit cone, and $\cong$ is thus not an arbitrary natural isomorphism. However, this property of $\cong$ in the first displayed formula mentioned is never stated on pages 99 nor 100. Am I right with this? Also this more general notion of $\cong$ is possible when weighted limit is considered instead of the usual one, even if $W$ is equal to *,and thus the weighted limit weighted by * may have other $\cong$ then the usual limit? Or Am I completely wrong and the $\cong$ is always unique?

Answer 1

All you need here is Yoneda. The limit $\ell$ of $F$ weighted by $W$ is a representation of the functor of $W$-weighted cones; the representation is uniquely determined by where it sends the identity map of the limit, which is a particular natural transformation from $W$ to $\mathrm{Hom}(\ell,F)$. When $W$ is the terminal functor and you're calculating an ordinary limit, this is a point of the latter homs, which is to say a natural transformation from $\ell$ to $F$, that is, the usual limit cone over $F$ you're looking for. Thus the isomorphism is here, and was in the case of ordinary limits, an arbitrary natural isomorphism.