I am reading "Differentiable Manifolds: A Theoretical Physics Approach" by Torres del Castillo. On page 116 of the book the author gives the equations for a parallel vector field along a curve $C$ as follows:
$$\frac{d(Y^k \circ C)}{dt} + \frac{d(x^i \circ C)}{dt} (\Gamma_{ji}^k \circ C)(Y^j \circ C) = 0$$
Then defines geodesics as the curves on which $C'$ is parallel to itself along C and gives the following equation:
$$\frac{d^2(x ^k \circ C)}{dt^2} +(\Gamma_{ji}^k \circ C) \frac{d(x^i \circ C)}{dt} \frac{d(x^j \circ C)}{dt} = 0$$
since we have that the coefficients of $C'$ are $Y^j = \frac{d(x^j \circ C)}{dt}$
But in my head, for example the first term of the first equation would become
$$\frac{d^2(x ^k \circ C \circ C)}{dt^2}\neq \frac{d^2(x ^k \circ C)}{dt^2}$$
which doesn't really makes sense. So how does this follow?