Take a topologically enriched small category $\mathcal{P}$ and the category of enriched diagrams of spaces $[\mathcal{P},\mathrm{Top}]_0$. We work with $\Delta$-generated spaces. 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.