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.