Let $I$ be a filtered category. We say a full subcategory $J$ is cofinal if for every object $A \in I$, there is an object $B \in J$ so that $\text{Hom}(A, B) \neq \emptyset$. Let $F : I \to \mathcal C$ be a functor from a filtered category admitting a colimit. Let $\iota : J \to I$ be the embedding of a cofinal subcategory into $I$. Show that $\text{colim} F \cong \text{colim}(F \circ \iota)$.
I realize that there is a similar post here, but reading about the answer and the question, I deduced that that post's question is slightly different from my question and that I couldn't parse enough of that post's answer.
Here is what I have so far:
Assume first that $\text{colim}(F \circ \iota)$ exists. Then I have proven just from the definition of a colimit that there is a unique morphism from $\text{colim}(F \circ \iota)$ to $\text{colim} F$.
The other direction is where I was having trouble with, because I was getting all confused with all the arrows and still haven't figured it out. Nonetheless here is my attempt.
Now, let $i_1 \xrightarrow{\theta} i_2$ be a morphism in $I$. Then there exists $j_1$ and $j_2$ in $J$ with $i_1 \xrightarrow{\alpha_1} j_1, i_2 \xrightarrow{\alpha_2} j_2$. Since $I$ is filtered and $J \subset I$, we have there there are some objects $d_1, d_2$ in $I$ such that there are maps $j_1 \xrightarrow{\beta_{j_1}} d_1, i_1 \xrightarrow{\beta_{i_1}} d_1, j_2 \xrightarrow{\beta_{j_2}} d_2, i_2 \xrightarrow{\beta_{i_2}} d_2$. Because there are two morphisms $\beta_{j_1} \circ \alpha_1$ and $\beta_{i_1}$ from $i_1$ to $d_1$, the filteredness of the category $I$ implies that there exists some morphism $d_1 \xrightarrow{\gamma_1} e_1$ for some $e_1 \in I$ such that $\gamma_1 \circ \beta_{j_1} \circ \alpha_1 = \gamma_1 \circ \beta_{i_1}$. Similarly, there is a morphism $d_2 \xrightarrow{\gamma_2} e_2$ for some $e_2 \in I$ such that $\gamma_2 \circ \beta_{j_2} \circ \alpha_2 = \gamma_2 \circ \beta_{i_2}$. Also, by definition of $\text{colim} F$, there are morphisms $e_1 \xrightarrow{\lambda_{e_1}} \text{colim} F$ and $e_2 \xrightarrow{\lambda_{e_2}} \text{colim} F$. Thus, we have the map $\lambda_{e_1} \circ \gamma_1 \circ \beta_{j_1}$ from $j_1$ to $\text{colim} F$ and $\lambda_{e_2} \circ \gamma_2 \circ \beta_{j_2}$ from $j_2$ to $\text{colim} F$. Yet because $j_1, j_2$ are in $J$, there are morphisms $j_1 \xrightarrow{\delta_{j_1}} \text{colim} F \circ \iota$ and $j_2 \xrightarrow{\delta_{j_2}} \text{colim} F \circ \iota$. Then by the definition of $\text{colim} F \circ \iota$ there is a unique map $f$ from $\text{colim} F \circ \iota$ to $\text{colim} $f. Here is the diagram:
I know that this is not what we are looking for, since we actually want a morhpism from $\text{colim} F$ to $\text{colim} F \circ \iota$ and not the other way around as is the case in the above diagram. However, I don't see how I can get such a morhpism, in such a way that 'everything commutes.' I also don't know how we can prove the final result without assuming that $\text{colim}(F \circ \iota)$ exists. Can you help?
If you don't know yet that $\mathrm{colim}(F\circ \iota)$ exists, there is really only one option, namely to show that $\mathrm{colim}\,F$ together with the appropriate structure morphisms satisfies the universal property that $\mathrm{colim}(F\circ\iota)$ has. Once you show this, you are done. (The natural comparison map $\mathrm{colim}(F\circ\iota)\to\mathrm{colim}\,F$ that you already obtained will then be an isomorphism as well.)
So, given a cocone $(\varphi_j\colon F\iota(j)\to T)_{j\in J}$ under $F\circ\iota$ in $\mathcal{C}$, you have to produce a map $\mathrm{colim}\,F\to T$ in $\mathcal{C}$ making the necessary triangles commute, and prove this map is unique.
Since you only have the universal property of $\mathrm{colim}\,F$ to work with, you have to extend the cocone $(\varphi_j)_{j\in J}$ to a cocone $(\psi_i\colon Fi\to T)_{i\in I}$ under $F$ such that $\varphi_j=\psi_{\iota(j)}$. To do so, there is again only really one thing you can do: given an arbitrary $i\in I$, choose a $j\in J$ and a map $i\to\iota(j)$, and define $\psi_i\colon Fi\to F\iota(j)\xrightarrow{\varphi_j} T$. You can now use your hypotheses of filteredness of $I$ and cofinality of $J$ (and that $J$ is a full subcategory) to show that this definition is independent of the choice of $j$ and the map $i\to\iota(j)$. Now, using this extended cocone, you get a map $\mathrm{colim}\,F\to T$ that makes the appropriate triangles commute. You can show uniqueness of this map given a certain cocone under $F\circ\iota$ in a similar way, by first extending the cocone to a cocone under $F$, and then using the uniqueness part of the universal property of $\mathrm{colim}\,F$.