Every weighted limit can be constructed from conical limits and cotensors. However, yesterday, a friend of mine, asked a question that may be rephrased as follows.

What is the reason that in the world of $\mathbf{Set}$-enriched categories every weighted limit can be constructed from conical limits (and trivial cotensors with $1$), and in the world of $\mathbf{Cat}$-enriched categories every weighted limit can be constructed from conical limits and cotensors with $2$?

Is it directly related to the fact that every set can be built upon $1$ and every category can be built upon $2$?

Is it possible to generalise these results to arbitrary (sufficiently well-behaved) monoidal category? For example, let us say that a symmetric monoidal closed category $\mathbb{V}$ is cocomplete and there exists a set $F$ of objects from $\mathbb{V}$ such that every object in $\mathbb{V}$ is a colimit of some objects from $F$. Is it true that every $\mathbb{V}$-weighted limit can be expressed via conical limits and cotensors with objects from $F$?