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.

Answer 2

Here's my two cents. We think of a little more general situation:

Let $\mathsf{I,J,C}$ be categories, and let $F:\mathsf{I}\to \mathsf{C},\,G:\mathsf{J}\to\mathsf{C}$ be any functros. Let $\alpha:F\Rightarrow G$ be a natural transformation, where we think of $F$ and $G$ as bifunctors $\mathsf{I}\times\mathsf{J}\to\mathsf{C}$ by precomposing projections $\mathsf{I}\times\mathsf{J}\to\mathsf{I,J}$. Then $\alpha$ induces a map $$\operatorname{colim}_\mathsf{I}F\to\operatorname{lim}_\mathsf{J}G,$$ provided that the displayed limit and colimit exist.

In our case, we take $Fi=\lim_{j\in\mathsf{J}}H(i,j)$ and $Gj=\operatorname{colim}_{i\in\mathsf{I}}H(i,j)$, with $\alpha=\iota\pi$.