colim functor preserves naturality

Let \(\mathcal{C}\) be a (sufficiently) cocomplete category and \(\mathcal{I}, \mathcal{J}\) be indexing categories
Suppose

\begin{align*} \mathcal{F}, \mathcal{G}: \mathcal{J} \times \mathcal{I} \rightarrow \mathcal{C} \end{align*}

are functors and \(\eta: \mathcal{F} \rightarrow \mathcal{G}\) a natural transformation

Denote for \(j \in J\)

\begin{align*} \mathrm{colim} \mathcal{F}(j,-), \mathrm{colim} \mathcal{G}(j,-), \end{align*}

as colimit of the induced diargams \(\mathcal{F}(j,-), \mathcal{G}(j,-) :I \rightarrow \mathcal{C}\) and \(\mathrm{colim}(\eta(j)): \mathrm{colim}(\mathcal{F}(j,-)) \rightarrow \mathrm{colim}(\mathcal{G}(j))\) the induced morphism.

Then \(\mathrm{colim}(\eta(j))\) is natural with respect to morphism in \(\mathcal{J}\)

follows from

• natural transformation as functor from the product with the interval category
  
• category cat as cartesian closed category
  

where we get a functor

\begin{align*} (\mathcal{F}, \mathcal{G}, \eta): \Delta^1 \times \mathcal{J} \times \mathcal{I} \rightarrow& \mathcal{C} \end{align*}

which under the adjunction corresponds to a functor

\begin{align*} \Delta^1 \times \mathcal{J} \rightarrow \mathrm{Fun}(\mathcal{I}, \mathcal{C}) \end{align*}

where postcomposition with the colim functor gives

\begin{align*} \Delta^1 \times \mathcal{J} \rightarrow \mathcal{C} \end{align*}

showing that the morphism induced by \(\Delta^1\) is natural with respect to \(\mathcal{J}\)