Let's consider a diagram $\Phi: \lambda \to \mathcal{T}_*$ $$ X_0 \to X_1 \to \cdots \to X_\xi \to X_{\xi+1} \to \cdots $$ of pointed spaces, indexed by some ordinal $\lambda$, in which each $X_\xi$ is a subspace of a certain space $Y$ and each map $X_\zeta\to X_\xi$ is the inclusion map.
I have the idea that 'morally' the categorical colimit of the diagram is simply the union $X = \bigcup_{\xi< \lambda} X_\xi$ (with the subspace topology). Certainly this is not actually the case in general -- we need some topological restrictions.
I'm willing to impose really strong topological restrictions. First of all, I'm happy to assume all of the maps $X_\zeta \to X_{\xi}$ and $X_\xi \to Y$ are cofibrations. I'd even be pretty happy to have an argument in which each map $X_\xi \to X_{\xi + 1}$ is obtained by attaching a cell, or a cone. Furthermore, I'm content to work with a category of spaces (such as CGWH) for which cofibrations are necessarily inclusions of closed subspaces (up to homeomorphism).
In general, of course, there is a comparison map $c: \mathrm{colim}\ \Phi\to X$, and it is clearly surjective. It would suffice, therefore, to prove that $c$ is a cofibration.
Unfortunately, my diagram-wrangling skills are coming up short. I'd appreciate pointers to a good argument or a useful reference.