Verdier-duality identification intertwines Yoneda product and oriented convolution

Typed correspondences. Put $t=r+d_j-d_i$ and $w=s+d_k-d_j$. Under the chain of [F3] the maps $\alpha$ and $\beta$ correspond to kernel forms $\kappa :k_A\to q^{!}{k}_j[t]$ and $\lambda :k_B\to v^{!}{k}_k[w]$, equivalently to cohomological correspondences $U:q_{!}{k}_A\to k_j[t]$ and $V:v_{!}{k}_B\to k_k[w]$; here $k_j=k_{X_j}$ and $k_k=k_{X_k}$. The middle square $(a,b;q,u)$ is Cartesian with $qa=ub$, and the three-factor identities $pa=\mathrm{\ell }h$, $vb=mh$ hold by construction of the triple fiber product.

The explicit composite. Let $\mathrm{B}\mathrm{C}:b_{!}{a}^{\ast }\stackrel{\sim }{\to }{u}^{\ast }{q}_{!}$ be the proper base change isomorphism of [F4]. Form the map $W:(vb{)}_{!}{k}_T=v_{!}{b}_{!}{a}^{\ast }{k}_A\stackrel{v_{!}\mathrm{B}\mathrm{C}}{\to }{v}_{!}{u}^{\ast }{q}_{!}{k}_A\stackrel{v_{!}{u}^{\ast }U}{\to }{v}_{!}{k}_B[t]\stackrel{V[t]}{\to }{k}_k[t+w]\mathrm{.}$ Its adjoint $k_T\to (vb{)}^{!}{k}_k[t+w]$ under $(vb{)}_{!}⊣(vb{)}^{!}$ is the composite $k_T\stackrel{a^{\ast }\kappa }{\to }{a}^{\ast }{q}^{!}{k}_j[t]\to b^{!}{u}^{\ast }{k}_j[t]=b^{!}{k}_B[t]\stackrel{b^{!}\lambda [t]}{\to }(vb{)}^{!}{k}_k[t+w]$, in which the middle arrow is the mate exchange of $\mathrm{B}\mathrm{C}$. The equality follows by applying the adjunction $(vb{)}_{!}⊣(vb{)}^{!}$ to both maps: the defining evaluation of the mate reproduces exactly the three arrows of $W$, so no exchange isomorphism beyond $\mathrm{B}\mathrm{C}$ is assumed.

Geometric interpretation of the mate. Put $S=A\times B$, $F=(q,u):S\to X_j\times X_j$, and $\theta :T\hookrightarrow S$. By [F1] the pulled-back positive purity counit is $g_F:{\theta }_{\ast }{k}_T[-2d_j]\to k_S$. The external dualizing class corresponding to $\kappa ⊠\lambda$ has degree $n+n^{\mathrm{\prime }}$, where $n=d_i+d_j-r$ and $n^{\mathrm{\prime }}=d_j+d_k-s$ by [F3]; composing it with $g_F$ and taking the adjoint under ${\theta }_{\ast }⊣{\theta }^{!}$ gives a class of degree $n+n^{\mathrm{\prime }}-2d_j$ on $T$. The graph-factorization and composition-of-counits comparison proved in [F1] identifies this supported cap map with the mate in step 2.1, followed by $b^{!}\lambda [t]$. Thus its representative is exactly the adjoint of $W$. The purity counit, its pullback, and the mate are fixed global morphisms; the positive normal-ball generator fixes their orientation, while their equality on arbitrary supports follows from that supported adjunction comparison.

Pushing the correspondence to the third corner. Its adjoint representative is $U_C:m_{!}{k}_C\stackrel{m_{!}{\eta }_h}{\to }{m}_{!}{h}_{\ast }{k}_T=(vb{)}_{!}{k}_T\stackrel{W}{\to }{k}_k[t+w]$, where ${\eta }_h:k_C\to h_{\ast }{h}^{\ast }{k}_C=h_{\ast }{k}_T$ is the ordinary adjunction unit and $h_{\ast }=h_{!}$ by [F4]. In kernel form this is $k_C\to h_{\ast }{k}_T\to h_{\ast }{h}^{!}{m}^{!}{k}_k[t+w]\stackrel{{ϵ}_h}{\to }{m}^{!}{k}_k[t+w]$ with ${ϵ}_h$ the dualizing proper trace, and dualizing transitivity identifies the middle dualizing object with ${\omega }_T[-2d_k+t+w]$. Step 3.1 therefore identifies the class on $C$ represented by $U_C$ with $h_{\ast }{\mathrm{\Delta }}_{X_j}^{!}({\mathrm{\Phi }}_{ij}^r(\alpha )⊠{\mathrm{\Phi }}_{jk}^s(\beta ))$, which by [F2] is the convolution product ${\mathrm{\Phi }}_{ij}^r(\alpha )\ast {\mathrm{\Phi }}_{jk}^s(\beta )$.

The composition square commutes. The inverse of the chain of [F3] sends $U$ to $f_{i!}{k}_i\stackrel{f_{i!}{\eta }_p}{\to }{f}_{i!}{p}_{\ast }{k}_A=f_{j!}{q}_{!}{k}_A\stackrel{f_{j!}U}{\to }{f}_{j!}{k}_j[t]$ and $V$ to the analogous map built from $u,v$, where ${\eta }_p:k_i\to p_{\ast }{p}^{\ast }{k}_i=p_{\ast }{k}_A$ is the adjunction unit and $f_{j!}=f_{j\ast }$. In the composite of these two maps, naturality of ${\eta }_u:\mathrm{id}\to u_{\ast }{u}^{\ast }$ moves $f_{j!}U$ past the second unit, and proper base change $u^{\ast }{q}_{!}\cong b_{!}{a}^{\ast }$ identifies the resulting two units with the unit ${\eta }_{pa}:k_i\to (pa{)}_{\ast }{k}_T$; the Beck–Chevalley unit identity is the transpose of the triangle equations under $f_{i!}⊣f_i^{!}$ and $(vb{)}_{!}⊣(vb{)}^{!}$, so both routes give the same morphism $f_{i!}{k}_i\stackrel{f_{i!}{\eta }_{pa}}{\to }{f}_{k!}(vb{)}_{!}{k}_T\stackrel{f_{k!}W}{\to }{f}_{k!}{k}_k[t+w]$. Applying the inverse map to $U_C$ gives the same expression, because $pa=\mathrm{\ell }h$ and the composed-unit formula ${\eta }_{\mathrm{\ell }h}={\mathrm{\ell }}_{\ast }{\eta }_h{\mathrm{\ell }}^{\ast }\circ {\eta }_{\mathrm{\ell }}$ follows from the same triangle equations. Hence the product square commutes for arbitrary cohomological correspondences, before orientation or grading is applied; this is the compatibility of products asserted in the Statement.

Shifts, degrees and signs. Restore the object shifts: $t+w=r+s+d_k-d_i$. The corner degrees $d_i+d_j-r$ and $d_j+d_k-s$ add and lose $2d_j$ under the refined intersection, giving $d_i+d_k-r-s$, which is the corner degree of the output. At a finite approximation the middle factor has dimension $d_j+b$ and each incoming ordinary Borel–Moore degree acquires $2b$, so the output acquires exactly $2b$ and the stable equivariant degree is unchanged. Homogeneous maps compose as $V[t]\circ U$, hence as $\beta [r]\circ \alpha$ after restoring shifts, with no interchanged factors and no additional sign; the only possible sign in re-ordering the complex normal factors is $(-1{)}^{(2d_j)(2d_j)}=+1$, and external products follow the usual tensor Koszul convention with no permutation occurring here.

Units and the algebra statement. Let $\delta :X_i\to Z_{ii}$ be the diagonal, with $p\delta =q\delta ={\mathrm{id}}_{X_i}$. The cohomological correspondence for its fundamental class is $U_{\delta }:q_{!}{k}_{Z_{ii}}\to q_{!}{\delta }_{\ast }{k}_{X_i}=k_{X_i}$, induced by restriction $k_{Z_{ii}}\to {\delta }_{\ast }{k}_{X_i}$. Its inverse image under the chain [F3] is $f_{i!}{k}_{X_i}\to f_{i!}{p}_{\ast }{k}_{Z_{ii}}\to f_{i!}{k}_{X_i}$; restriction of the first unit to $\delta$ is the identity, so this composite is the identity. The adjoint kernel of $U_{\delta }$ is the proper trace of the positive fundamental class of $X_i$, since ${\delta }^{!}{q}^{!}{k}_{X_i}=k_{X_i}$. Hence ${\mathrm{\Phi }}_{ii}^0({\mathrm{id}}_{L_i})={\delta }_{\ast }[X_i]=[{\mathrm{\Delta }}_{X_i}]$, of corner degree zero. Step 5.1 transfers associativity and these units from shifted composition; the unit on the finite sum is ${\sum }_i[{\mathrm{\Delta }}_{X_i}]$. The cornerwise bijections of [F3] therefore give the claimed unital graded algebra isomorphism, with nonmatching corners zero by [F2]. AC is inherited through [A1].