So there are two constructions that are relevant here:
1. A functor BG->C extends to a colimit diagram colim(F):BG^\triangleright -> C, which looks like X\to X_hG.
2. Given any map f:X -> Y, there is a composite BG^\triangleright -> \Delta^1 -> C which exhibits the compatibility of the map with the trivial G-action on X.
My claim is that construction (1) applied to X with trivial G-action produces the same functor as construction (2) applied to X->X_hG