Can the interval category be expressed as a colimit? A category with two objects and no non-identity morphisms can be expressed as the colimit of a diagram consisting of two copies of the trivial category. Naively, I would think that the interval category might be the colimit (or 2-colimit) of a diagram consisting of two copies of the trivial category connected by the identity functor (call this colimit $C$). However, the definition of colimit requires that everything commute. So, we have two morphisms $0, 1 : \text{pt} \to C$, and a commutative triangle $0 \cong 1 \circ \text{id}$, which tells me that $0 \cong 1$, even though, for the interval category, there should merely be a natural transformation from $0$ to $1$.
So, is it even possible to express the interval category as a (co)limit, or is it something that we just have to assume is there? Is there a weaker sort of limit that would work? Is there a different universal property I should be looking at?

$\DeclareMathOperator*{\colim}{colim}$
You can obtain the category $[1]=\{0\to 1\}$ joining two copies of the terminal category $[0]=\{0\}$. A join is a colimit: given simplicial sets $X,Y$ you have to compute
$$
\colim_{[n]\to [p+ q+1]} X_p\times Y_q
$$ or, in a more elegant way, the convolution
$$
\int^{p,q} X_p\times Y_q \times \Delta(-,[p]\oplus [q])
$$ where $\oplus$ is the ordinal sum $[p]\oplus [q]:=[p+q+1]$.
