I would not call it acclimatization. First of all, depending on the kind of fungus, you have to submitt your biomass to sterillization to remove bacteria, molds and possibly naturely present brown- and white-rot fungi, ascomycetes atc. in the starting material. In the case of application of sepsis resistant strains of fungi derived, e.g. from Phanerochaete chrysosporium, you might potentionally omit step of sterilization.
The simplest method of biomass sterilization is short time boiling in distilled water, combined with UV irradiation of the surface of cooled material prior to its placing on a grown mycelium of the selected fungus.
Maybe use of ethylene- or propylene oxide on sterilization of biomass in a closed chamber could serve well, on condition of fast and complete removal of the gas after biomass disinfection.
You shall have also to keep the optimum temperature and humidity of the air and of the biomass during "biopulping" . The optimum concentration of the air oxigen in the bioreactor /thermostat.../ has to be maintained, as well. These parameters may differ from one to another, according to the fungus applied.
The biopulping effect of certain fungi on the biomas may change within some limits from lignin-selective to errosive and vice versa, due to the substrate /soft wood-hard wood, grass..../ degraded.
Note: in the case of fungal treatment of biomass submitted to "steem exsplosion process" a problem of sterillization can be neglected.
It strikes me, sometimes when using errosive strains /Trametes versicolor, Phanerochaete chrysosporium/ of white-rot fungi, a small amount of D-glucose to water prior to biomass sterillization can be added. The aim is to supress the fungi production of cellulolytic enzymes.
When smaller dimensions biomass particles are used for biopulping, the suspension of an inoculum may be sprayed /sprinkled/ directly onto them. After homogenzation the inoculated material is transported into the bioreactor.