$\displaystyle \operatorname{softmax}_{\operatorname{dog}}\left(f_{\operatorname{dog}}(x), f_{\mathrm{cat}}(x)\right)=\frac{\exp \left(f_{\operatorname{dog}}(x)\right)}{\exp \left(f_{\operatorname{dog}}(x)\right)+\exp \left(f_{\mathrm{cat}}(x)\right)}$