Degree as an intersection with a regular value

Regularity makes $F$ transverse to $\{y\}$. Its finite signed count is the sum in [F1], since the local intersection signs against the positive point equal $\mathrm{sgn}(d{F}_p)$ by [F2]. The sum is $\mathrm{deg}(F)$; with compact $M$ the definition in [F2] names it $I(F,\{y\})$. Properness supplies finiteness even when $M$ is noncompact, but it does not enlarge that compact-source definition.

Assume $M$ compact. At $(p,y)$ a fibre tangent vector is $(u,0)$ and a graph tangent vector is $(v,d{F}_{p}v)$. The ordered derivative matrix for fibre first, graph second is $\left(\begin{array}{cc}I& I\\ 0& d{F}_p\end{array}\right)$, whose determinant has sign $\mathrm{sgn}(d{F}_p)$ in the induced orientations. The determinant-line calculation also handles $n=0$: the two source point signs from $M$ cancel, leaving the ambient point sign of $M\times N$, equal to $\mathrm{sgn}(d{F}_p)$. The transverse intersection is exactly the finite regular fibre, so summing gives $I(M\times \{y\},{\mathrm{\Gamma }}_F)=\mathrm{deg}(F)$. Reversing the two $n$-blocks multiplies each sign by $(-1{)}^{n^2}=(-1{)}^n$.

Write $s(p)=(p,F(p))$ and let $i$ include $M\times \{y\}$. Their oriented parametrizations identify $I(s,i)$ with the graph-first count, so [F4] gives $I(s,i)=(-1{)}^{n}I(s\times i,{\mathrm{\Delta }}_{M\times N})$. This agrees with the opposite-order graph sign in 2.1, establishing compatibility of the degree and diagonal conventions.