In the book "Lecture notes in algebraic topology" by Davis and Kirk I came across the following colimit $$E^{\infty} = \operatorname{colim}_{r \to \infty}E^r$$
where each of the $E^r$ were $R$-modules and $E^{\infty}$ was the $R$-module defined to be the colimit of the $E^r$'s
Now from what I understand about colimits, colimits aren't defined for a sequence of objects in some category (as the above notation seems to suggest), they are defined instead for functors. In the above, it seems that the colimit in question is a sequential colimit.
Now I'm assuming that this collection of $R$-modules gives rise to some functor $T$ from $(\mathbb{N}, \leq)$ (which is the filtered index category on the directed set $(\mathbb{N}, \leq)$) to $\operatorname{R-Mod}$, for which $T(n) = E^n$ and for which $T$ acts somehow on morphisms. Then I'm assuming that it will turn out that $\operatorname{co}\varinjlim T = E^{\infty} $ and we'd just label $\operatorname{colim}_{r \to \infty}E^r$ as $\operatorname{co}\varinjlim T$. Am I correct in saying this? If so let me move onto my next question.
The problem is that I don't know actually how to define such a functor $T$ on morphisms. The only possible morphisms on the $E^r$'s in the above case are endomorphisms $d^r : E^r \to E^r$.
Let me now restate this next question to be as self-contained as possible. Suppose that I have a collection of $R$-modules, $E^r$ for each $r \in \mathbb{N}$. How can I define a functor $T$ from $(\mathbb{N}, \leq)$ to $\operatorname{R-Mod}$ such that $\operatorname{co}\varinjlim T$ is $\operatorname{colim}_{r \to \infty}E^r$?
To give some context because I'm sure my question possibly isn't well-formed, the colimit $E^{\infty} = \operatorname{colim}_{r \to \infty}E^r$ is the colimit of the pages/sheets of a spectral seqeunce. See below for an excerpt of the passage where the colimit appears 
That notion is the (directed) colimit of the diagram $$E_0 \longrightarrow E_1 \longrightarrow E_2 \longrightarrow E_3 \longrightarrow \dots,$$ for some morphisms $\varphi_{ij} \colon E_i \rightarrow E_j$ such that $\varphi_{ik} = \varphi_{jk} \circ \varphi_{ij}$ for all $i \leq j \leq k$ and $\varphi_{ii} \colon E_i \rightarrow E_i$ is the identity. That is an object $E$ together with morphisms $\psi_i \colon E_i \rightarrow E$ satisfying $\psi_i = \psi_j \circ \varphi_{ij}$ whenever $i \leq j$ which is universal in the following sense:
For each obejct $F$ together with morphisms $\eta_i \colon E_i \rightarrow F$ satisfying $\eta_i = \eta_j \circ \varphi_{ij}$ whenever $i \leq j$ there exists exactly one morphism $f \colon E \rightarrow F$ such that the diagram
commutes for all $i,j$. The notion which is most often used is $E = \varinjlim E_i$, where the morphisms etc. are dropped.
In the case of modules the object $E$ is realized as a quotient of the direct sum of the $E_i$ and can be thought of as "gluing" the $E_i$ together according to the given maps.
We can also regard $\mathbb{N}$ as a directed index category and then the functor $\mathcal{F} \colon \mathbb{N} \rightarrow R-\text{mod}$ defining the $E_i$ and the $\varphi_{ij}$ as you mentioned.
As Eric stated as well it is not clear what the morphisms between the $E_i$ are supposed to be in your situation (at least for me).