The thing is that a product \$$\times\$$ isn't just a single object, it's a single object _and a pair of projection maps from that object to the original pair_.

Dually the coproduct \$$+\$$ is a single object _and a pair of inclusion maps from the original pair to that object_.

It's the fact that these projection/inclusion maps point in different directions that makes one a right adjoint and the other a left adjoint.