The existence of that sort of morphism between limits relates to some properties of the functor $g : I \rightarrow J$.

Given functors $h: I \rightarrow D$, $l: J \rightarrow D$ and a natural transformation $t : h \rightarrow lg$ we want to construct a morphism $Lim(h) \rightarrow Lim(l)$. Let $z = Lim(h)$, and let $s : z \rightarrow h$ be the universal cone. Suppose that for each object $x$ of $J$ a choice of an object $g^{-1}(x)$ in its fiber over $g$ is made. Then, for any $x$ we can define a morphism $z \rightarrow l(x)$ by $$z \xrightarrow{s_{g^{-1}(x)}} h(g^{-1}(x)) \xrightarrow{t_{g^{-1}(x)}} lg(g^{-1}(x)) = l(x)$$ If this family is a cone $z \rightarrow l$, then we get a morphism $z \rightarrow Lim(l)$ that we want. This family is a cone for example when for each arrow $f :x \rightarrow x'$ in $J$ there is a chain of arrows in $I$ $$g^{-1}(x) \xrightarrow{k_1} \xleftarrow{k'_1}\xrightarrow{k_2}...\xrightarrow{k_{n}}\xleftarrow{k'_n} g^{-1}(x')$$

such that $g(k'_i) = 1$ and $g(k_n)...g(k_2)g(k_1) = f$. For then we have $l(f)s_{g^{-1}(x)}t_{g^{-1}(x)} = s_{g^{-1}(x)}t_{g^{-1}(x)}$. For example for $n = 1$, omitting indexes for $s$ and $t$, this is shown by $$l(f)ts = lg(k_1)ts = th(k_1)s = ts = th(k'_1)s = lg(k'_1)ts = ts.$$

In your example you take $g^{-1}(M) = M$, $g^{-1}(R) = R$, and for $g^{-1}(S)$ you can take either $N$, $P$ or $Q$. Pretty much this is why the morphism between the limits exists.

The existence of that sort of morphism between limits relates to some properties of the functor $g : I \rightarrow J$.

Given functors $h: I \rightarrow D$, $l: J \rightarrow D$ and a natural transformation $t : h \rightarrow lg$ we want to construct a morphism $Lim(h) \rightarrow Lim(l)$. Let $z = Lim(h)$, and let $s : z \rightarrow h$ be the universal cone. Suppose that for each object $x$ of $J$ a choice of an object $g^{-1}(x)$ in its fiber over $g$ is made. Then, for any $x$ we can define a morphism $z \rightarrow l(x)$ by $$z \xrightarrow{s_{g^{-1}(x)}} h(g^{-1}(x)) \xrightarrow{t_{g^{-1}(x)}} lg(g^{-1}(x)) = l(x)$$ If this family is a cone $z \rightarrow l$, then we get a morphism $z \rightarrow Lim(l)$ that we want. This family is a cone for example when for each arrow $f :x \rightarrow x'$ in $J$ there is a chain of arrows in $I$ $$g^{-1}(x) \xrightarrow{k_1} \xleftarrow{k'_1}\xrightarrow{k_2}...\xrightarrow{k_{n}}\xleftarrow{k'_n} g^{-1}(x')$$

such that $g(k'_i) = 1$ and $g(k_n)...g(k_2)g(k_1) = f$. For then we can establish $l(f)s_{g^{-1}(x)}t_{g^{-1}(x)} = s_{g^{-1}(x)}t_{g^{-1}(x)}$. For example for $n = 1$, omitting indexes for $s$ and $t$, this is shown by $$l(f)ts = lg(k_1)ts = th(k_1)s = ts = th(k'_1)s = lg(k'_1)ts = ts.$$

In your example you take $g^{-1}(M) = M$, $g^{-1}(R) = R$, and for $g^{-1}(S)$ you can take either $N$, $P$ or $Q$. Pretty much this is why the morphism between the limits exists.

The existence of that sort of morphism between limits relates to some properties of the functor $g : I \rightarrow J$.

Given functors $h: I \rightarrow D$, $l: J \rightarrow D$ and a natural transformation $t : h \rightarrow lg$ we want to construct a morphism $Lim(h) \rightarrow Lim(l)$. Let $z = Lim(h)$, and let $s : z \rightarrow h$ be the universal cone. Suppose that for each object $x$ of $J$ a choice of an object $g^{-1}(x)$ in its fiber over $g$ is made. Then, for any $x$ we can define a morphism $z \rightarrow l(x)$ by $$z \xrightarrow{s_{g^{-1}(x)}} h(g^{-1}(x)) \xrightarrow{t_{g^{-1}(x)}} lg(g^{-1}(x)) = l(x)$$ If this family is a cone $z \rightarrow l$, then we get a morphism $z \rightarrow Lim(l)$ that we want. This family is a cone for example when for each arrow $f :x \rightarrow x'$ there is a chain of arrows in $I$ $$g^{-1}(x) \xrightarrow{k_1} \xleftarrow{k'_1}\xrightarrow{k_2}...\xrightarrow{k_{n}}\xleftarrow{k'_n} g^{-1}(x')$$

such that $g(k'_i) = 1$ and $g(k_n)...g(k_2)g(k_1) = f$. For then we have $l(f)s_{g^{-1}(x)}t_{g^{-1}(x)} = s_{g^{-1}(x)}t_{g^{-1}(x)}$. For example for $n = 1$, omitting indexes for $s$ and $t$, this is shown by $$l(f)ts = lg(k_1)ts = th(k_1)s = ts = th(k'_1)s = lg(k'_1)ts = ts.$$

In your example you take $g^{-1}(M) = M$, $g^{-1}(R) = R$, and for $g^{-1}(S)$ you can take either $N$, $P$ or $Q$. Pretty much this is why the morphism between the limits exists.

The existence of that sort of morphism between limits relates to some properties of the functor $g : I \rightarrow J$.

Given functors $h: I \rightarrow D$, $l: J \rightarrow D$ and a natural transformation $t : h \rightarrow lg$ we want to construct a morphism $Lim(h) \rightarrow Lim(l)$. Let $z = Lim(h)$, and let $s : z \rightarrow h$ be the universal cone. Suppose that for each object $x$ of $J$ a choice of an object $g^{-1}(x)$ in its fiber over $g$ is made. Then, for any $x$ we can define a morphism $z \rightarrow l(x)$ by $$z \xrightarrow{s_{g^{-1}(x)}} h(g^{-1}(x)) \xrightarrow{t_{g^{-1}(x)}} lg(g^{-1}(x)) = l(x)$$ If this family is a cone $z \rightarrow l$, then we get a morphism $z \rightarrow Lim(l)$ that we want. This family is a cone for example when for each arrow $f :x \rightarrow x'$ in $J$ there is a chain of arrows in $I$ $$g^{-1}(x) \xrightarrow{k_1} \xleftarrow{k'_1}\xrightarrow{k_2}...\xrightarrow{k_{n}}\xleftarrow{k'_n} g^{-1}(x')$$

such that $g(k'_i) = 1$ and $g(k_n)...g(k_2)g(k_1) = f$. For then we have $l(f)s_{g^{-1}(x)}t_{g^{-1}(x)} = s_{g^{-1}(x)}t_{g^{-1}(x)}$. For example for $n = 1$, omitting indexes for $s$ and $t$, this is shown by $$l(f)ts = lg(k_1)ts = th(k_1)s = ts = th(k'_1)s = lg(k'_1)ts = ts.$$

In your example you take $g^{-1}(M) = M$, $g^{-1}(R) = R$, and for $g^{-1}(S)$ you can take either $N$, $P$ or $Q$. Pretty much this is why the morphism between the limits exists.