Image of Colimit Between Mapping Spaces Let $D>0$ and $O$ be a non-empty, simply-connected, and open subset of $\mathbb{R}^D$.  For $d_n\geq d_m$, integers, define the injections
$$
\begin{aligned}
i^n_m: C(\mathbb{R}^{d_n};O) & \hookrightarrow C(\mathbb{R}^{d_m};O)\\
&f \mapsto f \circ \pi^n_m,
\end{aligned}
$$
where $\pi^n_m(x_1,\dots,x_{d_m})\mapsto (x_1,\dots,x_{d_m},\underbrace{0,\dots,0}_{d_m-d_n\, \mbox{times}})$.  Endowed each of these mapping spaces with their respective compact-open topologies; thus these maps become continuous.  
Then $\left(C(\mathbb{R}^{d_n};O),(i^n_m)_{n,m}\right)$ defines a filtered inductive system.  Then there "exists" a map:
$$
\begin{aligned}
i: \injlim C(\mathbb{R}^{d_n};O) &\rightarrow
C\left(\prod_{n \in \mathbb{N}} \mathbb{R}^{d_n};O\right)...
\end{aligned}
$$
But explicitly what is that map and what is it's image?  (Since I believe that in this case the right-hand side fails to be the topological colimit itself...)
 A: Let us consider more generally an inverse system $\mathbf X = (X_n,p_n)$ of spaces $X_n$ and bondings $p_n : X_{n+1} \to X_n$ and a space $Y$. Then we get induced maps 
$$p_n^* : C(X_n,Y) \to C(X_{n+1},Y), p_n^*(f) = f \circ p_n$$
and thus a direct system $C(\mathbf X,Y) = (C(X_n,Y),p_n^*)$.
Let $\projlim \mathbf X = (X,\pi_n)$ with projection maps $\pi_n : X \to X_n$. Then we get maps
$$\pi_n^* : C(X_n,Y) \to C(X,Y),  \pi_n^*(f) = f \circ \pi_n$$
which are compatible with the bondings of $C(\mathbf X,Y)$. In other words, we get a canonical map
$$j : \injlim C(\mathbf X,Y) \to C(X,Y) .$$
Now let us assume that all $p_n$ are surjections. Then also the $\pi_n$ are surjections. Thus the $p_n^*$ and the $\pi_n^*$ are injections. We conclude that also $j$ is an injection.
In your question we have an even more special case: The $p_n$ have left inverses $i_n : X_n \to X_{n+1}$. Then also the $\pi_n$ have left inverses: Define $\iota_n(x) = (x_k) \in X$ by $x_k = p_k \ldots p_{n-1}(x)$ for $k \le n$ and  $x_k = i_{k-1} \ldots i_n(x)$ for $k > n$. It is easy to see that this is thread in $\prod_{k=1}^\infty X_k$. 
This implies that the $p_n^*$ and the $\pi_n^*$ are embeddings. Thus the $C(X_n,Y)$ essentially form an ascending sequence of subspaces of $C(X,Y)$, thus their direct limit can be identified with the union of these subspaces although it may have a different topology than the subspace topology.
The map $j$ is in general not surjective. Its image is the set of all $\phi : X \to Y$ such that $\phi = f \circ \pi_n $ for some $n$ and $f \in C(X_n,Y)$. In your case we have $X  =\mathbb R^\infty$. With $Y = (-1,2)$ we can define $\phi((x_k)) = \sup_k \max(\lvert x_k \rvert, 1)$. This map does not have the above form.
