Let $\mathbf{X}$ vector field along curve $\gamma: [a,b] \longrightarrow \mathcal{M}$. We can parallel transporated each vector $\mathbf{X}(t)$ in the point $\gamma(a)$. That is, we can get a set of vectors $P_t^a \mathbf{X}(t)$ in the tangent space $T_{\gamma (a)} M$. I read in the book that this set can be viewed as a vector field along map $ P \circ \gamma : [a, b] \longrightarrow T_{\gamma (a)} M$.
I did not understand it. What exactly is a vector field? The map $P_t^a$? And what is the map P? And what is codomain? Codomain of vector field should be a tangent space of codomain map...
Sorry for my English
Here is this book, but there written on russian
A vector field is a smooth section $s:M\to TM$ of the tangent bundle. So if $\pi:TM\to M$ is the projection of the tangent bundle of $M$ onto $M$(takes each tangent space $T_pM$ to $p$), we have $\pi\circ s=id_M$.
Here is a reference on parallel transport.
Secondly, the codomain may be familiar to you as the range of or target space of a function. ( Though be careful: range of $f$ can mean image of $f$.)