I'd like to prove an analogous result for colimits.
I suppose the conclusion of the proposition for the limits should be that $\text{colim}:[\mathbf I,\mathscr A]\to\mathscr A$ defines a functor, and this functor is left adjoint to the diagonal functor.
I understand how to define the bijection $$\mathscr A(\text{colim} D, A)\simeq[\textbf I,\mathscr A](D,\Delta A)$$ and I managed to prove that it's natural in $A$.
But I don't understand how to prove naturality in $D$. Namely, suppose $\epsilon:D\to D'$ is a natural transformation. I don't understand how to define $\mathscr A(\text{colim} D, A)\to\mathscr A(\text{colim} D', A)$.
And I'm having the same kind of problem in defining the functor $\text{colim}$ on morphisms. If $\epsilon:D\to D'$ is a natural transformation, then I need to define $\text{colim }D\to\text{colim} D'$. I guess this has to be done using the universal property. So I need to show that $\text{colim} D'$ is a vertex of a cocone on $D$ and so there is a unique map from the colimit of $D'$ to this vertex. But I don't see how to get this cocone on $D$ with vertex $\text{colim} D'$.
Best Answer
For the second part of the question, a natural transformation $\epsilon : D \to D'$ defines a cocone on $D$ after composing with the colimit cocone on $D'$. Indeed a cocone on $D$ is a natural transformation from the functor $D$ to a constant functor on $\mathbf I$. The colimit cocone on $D'$, let us call it $\gamma$, is a natural transformation from $D'$ to the constant functor having values the colimit of the diagram $D'$. Hence $\gamma \circ \epsilon$ is a natural transformation from $D$ to the constant functor having value the colimit of $D'$. By universal property of the colimit cocone on $D$, we get a map from the colimit of $D$ to the colimit of $D'$.
In your drawing, the purple arrows are $\gamma \circ \epsilon$, you just need to add the arrows from $\gamma_i : D'(i) \to \mathrm{colim}D'$.
So now we have a map $\mathrm{colim}(\epsilon) : \mathrm{colim}D \to \mathrm{colim} D'$, this induces a natural transformation $\mathscr{A}(\mathrm{colim} D',-) \to \mathscr{A}(\mathrm{colim}D,-)$, often called $\mathrm{colim}(\epsilon)^*$.
This is just the general fact that $f : A \to A'$ induces a the natural transformation of covariant functors on $\mathscr{A}$, $f^* : \mathscr{A}(A',-) \to \mathscr{A}(A,-)$, (recall the (co)yoneda embedding $\mathscr{A}^{op} \to \mathbf{Fun}(\mathscr{A},\mathsf{Set})$).
The first variable in the hom functors is contravariant !