nLab
isocofibration

A morphism $f:A\to B$ in a strict 2-category $K$ is an isocofibration if the corresponding morphism $B\to A$ in $K^{\mathrm{op}}$ is an isofibration.

In $\mathrm{Cat}$, the isocofibrations are the functors which are injective on objects. There they are the cofibrations in the canonical model structure on Cat. However, a general 2-category has two candidates for a canonical model structure, one involving the isofibrations and one the isocofibrations, and in general the two may not be the same.