If we have two class and f_0 and f_1 as score for each class, then for cross entropy loss
eta_0 = SoftMax(f)[0] = exp(f_0)/(exp(f_0)+exp(f_1)) = 1/(1+exp(f_1-f_0))
hence, loss is equivalent to BCE with logit for single score (f_1-f_0)
If we have two class and f_0 and f_1 as score for each class, then for cross entropy loss
eta_0 = SoftMax(f)[0] = exp(f_0)/(exp(f_0)+exp(f_1)) = 1/(1+exp(f_1-f_0))
hence, loss is equivalent to BCE with logit for single score (f_1-f_0)