Left adjoint of pullback $Cat^1$-groups

Abstract

In [1] we define the pullback $Cat^1$-groups and showed that the category of pullback $Cat^1$-groups is equivalent to the category of pullback crossed modules. In this paper we proved that the pullback $Cat^1$-group has a left adjoint which is the induced $Cat^1$-group. We also give the left adjoint construction.