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.