Torsion tensor and affine connection symbols

I have read tons of questions about this topic but I think my particular issue is not solved. If so, please let me know.

So I want to prove that the torsion tensor $mathcal{T}$ actually transforms like a tensor. It is defined as $$mathcal{T}_{beta gamma}^{alpha}=Gamma_{beta gamma}^{alpha}-Gamma_{gamma beta}^{alpha},$$ where the $Gamma$ are the affine connection symbols.

In Relativity class we studied that the transformation of $Gamma$ between two reference systems $bar{alpha}$ and $alpha$ is given by

begin{equation}
Gamma_{bar{beta} bar{gamma}}^{bar{alpha}}=Gamma_{beta gamma}^{alpha} frac{partial x^{bar{alpha}}}{partial x^{alpha}} frac{partial x^{beta}}{partial x^{bar{beta}}} frac{partial x^{gamma}}{partial x^{bar{gamma}}}+frac{partial^{2} x^{alpha}}{partial x^{bar{beta}} partial x^{bar{gamma}}} frac{partial x^{bar{alpha}}}{partial x^{alpha}},
end{equation}

and it certainly leads me to $$ mathcal{T}_{bar{beta}bar{gamma}}^{bar{alpha}} = frac{partial x^{bar{alpha}}}{partial x^{alpha}} frac{partial x beta}{partial x^{bar{beta}}} frac{partial x^{gamma}}{partial x^{bar{gamma}}} mathcal{T}_{beta gamma}^{alpha},$$

which proves that the torsion tensor transforms as a tensor.

My question is: what would happen if the connection represented by $Gamma$ wasn’t an affine one? Would the transformation of $Gamma$ be different?