On the canonical map $\text{colim}_I \text{lim}_JH(i,j) \longrightarrow \text{lim}_J\text{colim}_IH(i,j)$

Question

I'm working through a proof of the existence of a canonical mapping

$$
\mu: \text{colim}_I \text{lim}_JH(i,j) \longrightarrow \text{lim}_J\text{colim}_IH(i,j) \tag{1}
$$
induced by a cone $(\mu_i: \text{lim}_JH(i,-)\to \text{lim}_J\text{colim}_I H(i,j))_{i \in I}$ with $\mu_i$ defined as the map induced by the cone $(\mu_{ij})_{j \in J}$ defined as
$$
\text{lim}_JH(i,-) \xrightarrow{\pi_j} H(i,j) \xrightarrow{\iota_i} \text{colim}_IH(-,j).
$$

I have yet to show that these are in effect cones: what approach can be taken in order to prove this in the least 'chaotic' way possible? My attempts so far have involved an amount of arrows that I can hardly keep track of, so maybe there is an elegant solution lurking in the corners which I am failing to see.

Answer 1

I think the shortest way is to use two ingredients :

1. the identification between functors $H:I\times J\to C$ and functors $I\to C^J$
2. the functoriality of (co)limits.

Indeed if you think of $H$ as a functor $I\to C^J$, taking its colimit gives you a cone $$H(i,j) \xrightarrow{\iota_i} \text{colim}_IH(-,j).$$ in the functor category $C^J$. Then you can apply the functor $\lim_J$ on this cone to directly obtain the cone $$(\mu_i: \text{lim}_JH(i,-)\to \text{lim}_J\text{colim}_I H(i,j))_{i \in I}$$ in the category $C$. From there you get the canonical mapping $\mu$, as you say in your question.