Take a topologically enriched small category $\mathcal{P}$ and the category of enriched diagrams of spaces $[\mathcal{P},\mathrm{Top}]\_0$. We work with the category of $\Delta$-generated spaces equipped with the mixed model structure. Suppose that the injective model structure exists (the paper <http://dx.doi.org/10.4310/HHA.2019.v21.n2.a15> gives some sufficient conditions).



> 
> Is there an explicit description of a fibrant replacement somewhere ?
> 
> 
> 


I can only understand that the injective fibrant diagrams are some kind of cofree enriched diagrams.


EDIT: by explicit, I mean which enables us to make some calculations.



Fibrant replacement of an injective model category of enriched diagrams
Section 8 of my paper [All (∞,1)-toposes have strict univalent universes](https://arxiv.org/abs/1904.07004) shows that under fairly general conditions, injective fibrant replacements can be given by cobar constructions (e.g. the dual of Corollary 8.16). I think this will apply to your situation if the hom-objects of $\mathcal{P}$ are cofibrant and the inclusions of identity morphisms are cofibrations. I don't know what sort of calculations you want to do, but cobar constructions come with a filtration that sometimes gives rise to calculational tools like spectral sequences.


