I do think that the pullback of differential geometry is also a pullback in the sense of category theory, however I did have some trouble completely justifying this thought. This is were I got:
Lets consider a vector field $\eta : Y \rightarrow T(Y)$ and a diffeomorphism $f: X \rightarrow Y$. Then from differential geometry the pullback can be written as, $$ f^*(\eta)(x) = Tf^{-1}\circ \eta \circ f(x).$$
Thus the commutative diagram becomes:
Now since $Tf^{-1}$ is invertible, we can use $Tf$ to write commutative diagram as a fiber product. From category theory we can complete the diagram of the vector field $\eta: Y \rightarrow T(Y)$ and the tangent map $Tf: T(X)\rightarrow T(Y)$ to obtain
Now due to the uniqueness of the fiber product up to isomorphism we see that $Y \times_{T(Y)} T(X)$ equals $X$ . Also $\eta^*(Tf)$ should equal $f$ (up to isomorphism). But then $f^*\eta$ should equal $Tf^*\eta$, however the notation $f^*\eta$ does not make much sense to me, is this just a short hand? If this isn't the case can someone give me somewhat more information about $ f^*\eta$ in perspective of category theory?
Thanks in advance.
The right way to think about this is that the answer to your question is no. It may be possible to write the pullback of a vector field along a diffeomorphism as a pullback in the sense of category theory, but this is basically a coincidence and is not normally a helpful way to understand the concepts.
The word "pullback" is used very broadly to describe, well, pulling things back. In other words, whenever you have a map $f:X\to Y$, and there is a natural way to turn some structure on $Y$ into a structure on $X$, you can describe that as a "pullback". Pullbacks in the sense of category theory are one example of such a thing (where the structure on $Y$ is a map from some object to $Y$), and pullbacks of vector fields are another (where the structure on $Y$ is a vector field; notably, though, you can only perform such a pullback if $f$ is an open immersion).