Let $C$ be a complete category, let $I$ be a small category, let $F,G:I\to C$ be functors, and let $W,U:C\to\mathrm{Set}$ be also functors, which we view as "weights".
The [weighted limits](https://ncatlab.org/nlab/show/weighted+limit) are then defined, $\lim\_W F$ and $\lim\_U G$.


Suppose now that we have natural transformations $\alpha:F\Rightarrow G$ and $\beta:W\Rightarrow U$.
Do $\alpha$ and $\beta$ induce a morphism $\lim\_W F\to\lim\_U G$, analogously to what happens with ordinary limits?



Functoriality of weighted limits
Let $X$ be an arbitrary object in $\mathcal{C}$.
I write $\{ W, F \}$ for the limit of $F$ weighted by $W$.
By definition,
$$\mathcal{C} (X, \{ W, F \}) \cong [\mathcal{I}, \textbf{Set}] (W, \mathcal{C} (X, F))$$
naturally in $X$.
If we have $U \Rightarrow W$ (note the direction!) and $F \Rightarrow G$ then functoriality of $W$ and $F$ on the RHS gives a map
$$[\mathcal{I}, \textbf{Set}] (W, \mathcal{C} (X, F)) \longrightarrow [\mathcal{I}, \textbf{Set}] (U, \mathcal{C} (X, G))$$
natural in $X$.
Applying the definition of weighted limit, this can be identified with a map
$$\mathcal{C} (X, \{ W, F \}) \longrightarrow \mathcal{C} (X, \{ U, G \})$$
natural in $X$.
Now apply the Yoneda lemma to obtain $\{ W, F \} \to \{ U, G \}$.
In short, $\{ W, F \}$ is *contravariantly* functorial in $W$ and covariantly functorial in $F$.


Incidentally, it is not necessary to hypothesise completeness – you have this morphism as soon as the weighted limits in question exist.


