Direct limit of the sequence $E_{0} \hookrightarrow E_{1} \hookrightarrow \cdots$ in the category of Banach spaces Recently I have been reading the paper The categorical origins of Lebesgue integration by Tom Leinster (https://arxiv.org/pdf/2011.00412.pdf). In this paper, he said that：

For $n \geq 0$, let $E_{n}$ be the subspace of $L^{p}[0,1], (1\leq p<\infty)$ consisting of the equivalence classes of step functions constant on each of the (open) intervals $\left(\frac{i-1}{2^{n}}, \frac{i}{2^{n}}\right)$, $\left(1 \leq i \leq 2^{n}\right)$. Write $E=\bigcup_{n \geq 0} E_{n}$, which is the space of step functions whose points of discontinuity are dyadic rationals. Then $E$ is dense in the set of all step functions on $[0,1]$, which in turn is dense in $L^{p}[0,1]$; so $E$ is dense in $L^{p}[0,1]$. It follows that $L^{p}[0,1]$ is the colimit (direct limit) of the diagram $E_{0} \hookrightarrow E_{1} \hookrightarrow \cdots$ in the category $\mathbf{\text{Ban}}$ of Banach spaces with linear contractions as morphisms.

I know that the category $\mathbf{\text{Ban}}$ is cocomplete, i.e., the colimit of any diagram exists (because it has coproducts and coequalizers). However, I just have no idea why the density of $E$ in $L^p[0,1]$ implies just that $L^p[0,1]$ is the colimit of the increasing sequence $E_{0} \hookrightarrow E_{1} \hookrightarrow \cdots$ in the category $\mathbf{\text{Ban}}$ of Banach spaces with linear contractions? I thought about this for two days but didn't figure out anything. Can anyone give a convincing or inspiring explanation?
Any help is appreciated.
 A: The density of $E$ in $L^p[0,1]$ means that $L^p[0,1]$ is the completion of $E$ as a (pseudo-)normed vector space. The relation with colimits has to do with how these completions function.
Banach spaces are a reflective subcategory of the category of (pseudo-)normed spaces: if $V$ is a (pseudo-)normed vector space, then it is equipped with a completion map $V\to\hat V$ for $\hat V$ a Banach space, such that any other map $V\to B$ with $B$ a Banach space factors uniquely as $V\to\hat V\to B$.
Consequently, any cocone $V_i\to B$ factors uniquely as $V_i\to(\mathrm{colim}_i V_i)\to B$ and then as $V_i\to\hat{\mathrm{colim}_i V_i}\to B$. Thus the completion of a colimit of Banach spaces considered as pseudo-normed vector spaces is the colimit of them considered as Banach spaces.
Now the union $E$ of $E_0\hookrightarrow E_1\hookrightarrow\cdots$ is the direct limit in the category of (pseudo-)normed spaces, so it being dense in $L^p[0,1]$ means $L^p[0,1]$ is the direct limit of $E_0\hookrightarrow E_1\hookrightarrow\cdots$ in the category of Banach spaces.
