Thanks Anindya and Keith for you answers and tips.

I think I have managed to figure out why the direction changes; it basically comes down to the fact that the way you count direct sums and products is different. So Christopher pointed out that for products :

\\[|S|^{|L_0(X)|} \ast |T|^{|L_1(X)|} = (|S|\ast|T|)^{|X|}\\] must be true so therefore \\[|S|^{X|} \ast |T|^{|X|} = (|S|\ast|T|)^{|X|}\\].

But for sums we need \\(|S|+|T|\\) and not \\(|S|\ast|T|\\). So the arrows need to be switched around so that everything counts up :

\\[|X|^{|S|} \ast |X|^{|T|} = |X|^{||S|+|T||}\\]

Now that I have that straight I am still trying to figure out Anindya's answer... Why is there a projection map for products and inclusion map for coproducts?

I think I have managed to figure out why the direction changes; it basically comes down to the fact that the way you count direct sums and products is different. So Christopher pointed out that for products :

\\[|S|^{|L_0(X)|} \ast |T|^{|L_1(X)|} = (|S|\ast|T|)^{|X|}\\] must be true so therefore \\[|S|^{X|} \ast |T|^{|X|} = (|S|\ast|T|)^{|X|}\\].

But for sums we need \\(|S|+|T|\\) and not \\(|S|\ast|T|\\). So the arrows need to be switched around so that everything counts up :

\\[|X|^{|S|} \ast |X|^{|T|} = |X|^{||S|+|T||}\\]

Now that I have that straight I am still trying to figure out Anindya's answer... Why is there a projection map for products and inclusion map for coproducts?