Let $D$ be a limit-complete category. My vague question is: given two diagrams in $D$, what comparisons between them induce a morphism of their limits? I'm especially interested in the case that the comparison has something to do with colimits.

Here is a well-known case in which the comparison has nothing to do with colimits. Let $I,J$ be small categories, and let $X:I\to D$ and $Y:J\to D$ be functors. Given a functor $f:J\to I$ and a natural transformation $\alpha:X\circ f\to Y$, there is an induced morphism between their limits $\lim(f,\alpha):\lim(X)\to \lim(Y)$

But today I came across another case in which one gets such a map between limits. Let $D={\bf Set}$ be the category of sets. Suppose $I$ is the W-shaped category $M\to N\leftarrow P\to Q\leftarrow R$ in $D$. Suppose $J$ is the cospan $M\to S\leftarrow R$, and let $g:I\to J$ be the functor sending $M\mapsto M, R\mapsto R$ and $N,P,Q\mapsto S$. \begin{align} I:\hspace{.5in}M\to \fbox{$N\leftarrow P\to Q$}\leftarrow R\\\\ g\downarrow\hspace{.8in}\\\\ J:\hspace{.5in}M\xrightarrow{\hspace{.5in}} S\xleftarrow{\hspace{.5in}} R \end{align} Given a functor $X:I\to D$, let $Y=Lan_gX:J\to D$ be the left Kan extension of $X$ along $g$. To my (perhaps very naive) surprise, I get a morphism of limits in this case too, $\lim(X)\to \lim(Y)$. To be explicit, this map sends a tuple $(m,p,r)\in \lim(X)$ to the pair $(m,r)\in \lim(Y)$.

**Question 1 (Sets)**: Is it true for any small categories $I, J$, and $X:I\to{\bf Set}$ that a functor $g:I\to J$ induces a function $\lim(X)\to \lim(Lan_gX)$?

**Question 2 (General)**: Is it true for general $D$ that for any categories $I, J$, and $X:I\to D$ that a functor $g:I\to J$ induces a morphism $\lim(X)\to \lim(Lan_gX)$?

**Question 3 (Vague)**: Is this part of any more general result about the interaction between limits and colimits?

Any references would be appreciated.