Working in compactly generated weak Hausdorff spaces, is the category of inclusion prespectra bicomplete?

I should probably specify that by inclusion prespectra, I mean prespectra such that the adjoint structure maps are closed inclusions. It is probably complete, as close inclusions of CGWH spaces are characterized by a limit condition. But is it cocomplete? It would be enough to define a left-adjoint to the forgetful functor to prespectra (or even injection prespectra), but I was not able to construct one.

no(simply because the "closed inclusion" condition seems too delicate to be preserved by various quotients), but I don't have a counterexample. $\endgroup$ – Tim Campion Oct 20 '18 at 19:43