The local intersection sign of graph against diagonal is sign det(I-Df)

The determinant at a fixed point. Let $x$ be a fixed point and let $(e_1,\dots ,e_n)$ be a positively oriented basis of $T_{x}M$. By [F2] the tuples $(e_i,D{f}_x{e}_i{)}_{i\le n}$ and $(e_j,e_j{)}_{j\le n}$ are positively oriented bases of $T_{{\gamma }_f(x)}{\mathrm{\Gamma }}_f$ and $T_{(x,x)}{\mathrm{\Delta }}_M$; concatenated in the order $({\mathrm{\Gamma }}_f,{\mathrm{\Delta }}_M)$ and expressed in the ambient basis $((e_1,0),\dots ,(e_n,0),(0,e_1),\dots ,(0,e_n))$ of $T_{(x,x)}(M\times M)$ they form the columns of the block matrix $\left(\begin{array}{cc}I& I\\ D{f}_x& I\end{array}\right)$. Subtracting the $i$-th column from the $(n+i)$-th column for each $i$ (a column shear of determinant $1$, The determinant is alternating and multilinear in the rows as well as in the columns applied to the transpose, using For every square matrix over a commutative ring, $\det (A^{\mathsf{T}})=\det (A)$) gives the block lower triangular matrix $\left(\begin{array}{cc}I& 0\\ D{f}_x& I-D{f}_x\end{array}\right)$, whose determinant is $\det (I-D{f}_x)$. The orientation comparison of [F1] is therefore $\mathrm{sign}\det (I-D{f}_x)$, and by [F3] this is ${\mathrm{ind}}_x(f)$.

The global intersection number. By [F3] the graph map is transverse to the diagonal precisely because all fixed points are nondegenerate, and transversality plus closedness of the complementary-dimensional submanifolds in the compact $M\times M$ makes the intersection finite; by [L1] the oriented intersection number $I({\gamma }_f,{\mathrm{\Delta }}_M)$ is the sum of the local signs over the fixed points, which by step 1.1 is ${\sum }_x{\mathrm{ind}}_x(f)=I(f)$, the geometric Lefschetz number of Geometric Lefschetz number (index sum). Replacing the graph map by the inclusion of the graph changes nothing: the two are identified by the diffeomorphism $x\mapsto (x,f(x))$, which is orientation-preserving and conjugates the local data, so $I({\mathrm{\Gamma }}_f,{\mathrm{\Delta }}_M)=I({\gamma }_f,{\mathrm{\Delta }}_M)$.

Factor order. In the opposite order $({\mathrm{\Delta }}_M,{\mathrm{\Gamma }}_f)$ the ambient tangent space is presented with the two $n$-dimensional factors exchanged, and [L1] applies with $a=b=n$, giving $I({\mathrm{\Delta }}_M,{\mathrm{\Gamma }}_f)=(-1{)}^{n^2}I({\mathrm{\Gamma }}_f,{\mathrm{\Delta }}_M)=(-1{)}^{n}I({\mathrm{\Gamma }}_f,{\mathrm{\Delta }}_M)$; the same factor appears pointwise because the local sign of [F1] is computed from the ordered sum. No metric is used, and the only choice principle involved is the Countable Choice recorded in [L1] for the intersection number of non-transverse representatives, which the transverse case of this lemma does not use.