On page $55$ of Do Carmo's Riemannian geometry, he proves that there is a unique symmetric affine connection compatible with a given metric on a manifold M. He defines it by a formula $\langle Z, \nabla_Y X \rangle = 1/2( C_{++-} ( X\langle Y, Z \rangle) + C_{-++}( \langle [X,Y], Z \rangle) )$ (where $C$ denotes a cyclic sum with altered signs (not his notation, I'm just being lazy)).
He then says that well-definedness of this definition can easily be checked.
I do not understand where there is even a question of well-definedness. No choices were made in the construction of this definition - or were they?
Thanks!
Perhaps the issue is to show that $\nabla_Y X$ is a tensor. That is, you need: $$ \langle \phi Z, \nabla_Y X \rangle = \phi \langle Z, \nabla_Y X \rangle $$ for any scalar field $\phi$. Or to put it another way, $ \langle Z, \nabla_Y X \rangle$ depends only on the pointwise value of $Z$, and not on any of its derivatives.
Then you will also need to show that $\nabla$ satisfies the axioms of a connection.