The following is an excerpt from volume II of Borceux:
Here $F$ is the left adjoint to the forgetful functor $U:\mathsf{Mod}_\mathcal{T}\longrightarrow \mathsf{Set}$. I'm confused and can't manage to actually show the "immediate" fact. If we take the "adjoints" of $u,v$ then we lose the coequalizer diagram.. Step-by-step assistance would be much appreciated!