$f: X \to Y$ and $Z$ are appropriate for intersection theory $X,Y,Z$ are boundaryless oriented manifolds, $X$ is compact, $Z$ is closed submanifold of $Y$, and $\dim X + \dim Z = \dim Y$), $f$ is transversal to $Z$. According to the text:
The orientation of $X$ provides an orientation of $df_xT_x(X).$ Then the orientation number at $x$ is $+1$ if the orientation on $df_xT_x(X)$ and $T_z(Z)$ "add up" to the prescribed orientation on $Y$. -- Guillemin and Pollack, Differential Topology Page 108
So I am totally confused. What does it mean by "add up" to the prescribed orientation? Does it simply mean first $df_xT_x(X)$ and then $T_z(Z)$ counterclockwisely?
Definition: Orientation of $V$, a finite-dimensional real vector space: Let $\beta, \beta^\prime$ be ordered basis of $V$, then there is a unique linear isomorphism $A: V \to V$ such that $\beta = A \beta^\prime$. The sign given an ordered basis $\beta$ is called its orientation.
Definition: Orientation of $X$, a manifold with boundary: A smooth choice of orientations for all the tangent space $T_x(X).$
What does counterclockwisely even mean in this context? You just take a postively oriented basis in $T_x(X)$, transport it by $df$, and then add to it a postively oriented basis of $Z$ and check whether or not they give a positively oriented basis of the tangent space. Or you could do it in the other order, it doesn't really matter, as long as you are consistent, only the sign will change.