Finite Deligne products exist via tensor-product algebras

By [F1] the tensor product $X{\otimes }_{k}Y$ carries commuting left actions of $R$ and $S$, hence the left $T$-action $(r\otimes s)(x\otimes y)=rx\otimes sy$ of the $k$-algebra $T=R{\otimes }_{k}S$ (Algebras over a commutative ring, central structure maps, and algebra homomorphisms), and it is functorial in both variables with the identity and composition laws [F2]; choosing a finite basis of $X$ identifies $X{\otimes }_{k}Y$ with a finite direct sum of copies of $Y$ by the universal property of [F2], so ${\dim }_k(X{\otimes }_{k}Y)={\dim }_{k}X\cdot {\dim }_{k}Y$ [F6] and $⊠$ lands in the finite-dimensional category $T\text{-}\mathrm{m}\mathrm{o}\mathrm{d}$ (Finite-dimensional vector space, and its dimension ${\dim }_{F}V$; infinite-dimensional means having no finite basis). Each partial functor is $k$-linear because additivity and scalar multiplication are read off on generators, where $(f+f^{\mathrm{\prime }})\otimes 1=f\otimes 1+f^{\mathrm{\prime }}\otimes 1$ and $(\lambda f)\otimes 1=\lambda (f\otimes 1)$, and the maps are determined by their values on generators [F2]; hence $(X,Y)\mapsto X{\otimes }_{k}Y$ is $k$-linear in each variable (k-linear categories and k-linear functors, Product category and its projection functors).

For fixed $Y$ the functor $X\mapsto X{\otimes }_{k}Y$ preserves cokernels: if $q:X\to X^{\mathrm{\prime }}$ is the cokernel of $f:X^{\mathrm{\prime }\mathrm{\prime }}\to X$, so that $\ker q=\mathrm{im}f$ (Exact sequences and short exact sequences of modules, Module homomorphism and isomorphism, kernel, image and cokernel), then $q\otimes 1$ is surjective with $\mathrm{im}(f\otimes 1)\subseteq \ker (q\otimes 1)$, and the balanced map sending $(x^{\mathrm{\prime }},y)$ with $q(x)=x^{\mathrm{\prime }}$ to the class of $x\otimes y$ is well defined because representatives differ by an element of $\ker q=\mathrm{im}f$, and it induces a two-sided inverse of the map on generators [F2]; hence $\ker (q\otimes 1)=\mathrm{im}(f\otimes 1)$. The functor is additive by the same generator computation, so it preserves finite coproducts and cokernels and therefore every finite colimit, that is, it is right exact (Left exact and right exact functors, An additive functor preserves finite biproducts); the argument in the first variable is identical, so $⊠$ is right exact in each variable.

Let $G,G^{\mathrm{\prime }}:T\text{-}\mathrm{m}\mathrm{o}\mathrm{d}\to \mathcal{E}$ be $k$-linear and right exact. Both $G\circ ⊠$ and $G^{\mathrm{\prime }}\circ ⊠$ are $k$-linear and right exact in each variable by step 2.1, so [F3] applies to the pair $(G\circ ⊠,G^{\mathrm{\prime }}\circ ⊠)$ and gives a bijection $\mathrm{Nat}(G\circ ⊠,G^{\mathrm{\prime }}\circ ⊠)\cong {\mathrm{Hom}}_T(G(T),G^{\mathrm{\prime }}(T))$ onto right $T$-module maps, by evaluation at $(R,S)$. Applying [F3] with the second algebra $k$ to the bifunctors $(Z,V)\mapsto G(Z{\otimes }_{k}V)$ and $(Z,V)\mapsto G^{\mathrm{\prime }}(Z{\otimes }_{k}V)$, whose value at $(T,k)$ is $G(T{\otimes }_{k}k)\cong G(T)$, gives in the same way a bijection $\mathrm{Nat}(G,G^{\mathrm{\prime }})\cong {\mathrm{Hom}}_T(G(T),G^{\mathrm{\prime }}(T))$: injectivity because the component at $(Z,k)$ recovers ${\eta }_Z$ under the natural isomorphism $Z{\otimes }_{k}k\cong Z$, and surjectivity because the extension of a right $T$-module map constructed by [F3] is natural and has the prescribed component. Since $(\eta \circ ⊠{)}_{R,S}={\eta }_T$, composition with $⊠$ identifies these two bijections, so restriction along $⊠$ is fully faithful.

Let $H$ be a $k$-linear bifunctor right exact in each variable and let $\bar{H}$ be the functor constructed from $W=H(R,S)$ in [F3]. Write $K$ for the functor on finite free $T$-modules used in that construction. Then $\bar{H}$ is $k$-linear: if $g_u,g_v$ are lifts of $u,v:Z\to Z^{\mathrm{\prime }}$ between the chosen presentations, then $g_u+g_v$ lifts $u+v$ and the induced map on cokernels is additive in the lift by the uniqueness in [F7], so $\bar{H}(u+v)=\bar{H}(u)+\bar{H}(v)$, and $g_{\lambda u}=\lambda g_u$ gives $\bar{H}(\lambda u)=\lambda \bar{H}(u)$ similarly. It is right exact: for a presentation $T^m\stackrel{\delta }{\to }{T}^n\to Z\to 0$ the universal property of the cokernel [F7] gives a bijection $\mathcal{E}(\bar{H}(Z),E^{\mathrm{\prime }})\cong \{h:W^n\to E^{\mathrm{\prime }}:hK(\delta )=0\}$ natural in $E^{\mathrm{\prime }}$, and the right-hand side is the set of $T$-linear maps $Z\to \mathcal{E}(W,E^{\mathrm{\prime }})$ under the left $T$-action $(t\cdot \phi )(w)=\phi (w\cdot t)$ on the hom-object, a map $h$ corresponding to the family of its components $h{i}_j:W\to E^{\mathrm{\prime }}$ and the condition $hK(\delta )=0$ saying exactly that this family annihilates $\delta (T^m)$; hence a cokernel sequence $Z^{\mathrm{\prime }\mathrm{\prime }}\to Z\to Z^{\mathrm{\prime }}\to 0$ in $T\text{-}\mathrm{m}\mathrm{o}\mathrm{d}$ gives exact sequences $0\to \mathcal{E}(\bar{H}(Z^{\mathrm{\prime }}),E^{\mathrm{\prime }})\to \mathcal{E}(\bar{H}(Z),E^{\mathrm{\prime }})\to \mathcal{E}(\bar{H}(Z^{\mathrm{\prime }\mathrm{\prime }}),E^{\mathrm{\prime }})$ for all $E^{\mathrm{\prime }}$, which by the characterization in [F7] says that $\bar{H}(Z^{\mathrm{\prime }\mathrm{\prime }})\to \bar{H}(Z)\to \bar{H}(Z^{\mathrm{\prime }})$ is again a cokernel sequence. So $\bar{H}$ preserves cokernels and, being additive, all finite colimits, that is, it is right exact (Left exact and right exact functors, An additive functor preserves finite biproducts, Abelian category). By [F3] there is a natural isomorphism $\bar{H}\circ ⊠\cong H$. Fix presentation data once on the small source. The construction on compatible action maps in the determination lemma then makes $H\mapsto \bar{H}$ a functor, with $\bar{H}\circ ⊠\cong H$ natural in $H$. Full faithfulness from step 3.1 lifts this comparison uniquely to $\overline{G\circ ⊠}\cong G$, natural in $G$, so restriction and this functor are quasi-inverse, giving an equivalence of categories (Equivalence, quasi-inverse, and adjoint equivalence of categories, Functor category $[\mathcal{C},\mathcal{D}]$, Natural transformation and its components, Class-sized category theory in ZFC: definable-class schemas, small and locally small categories, and why $\mathbf{C}\mathbf{A}\mathbf{T}$ is not formed).

By [F4] $T\text{-}\mathrm{m}\mathrm{o}\mathrm{d}$ is an abelian $k$-linear category, and because ${\dim }_{k}T={\dim }_{k}R\cdot {\dim }_{k}S$ is finite [F6] while the module category of a finite-dimensional $k$-algebra is intrinsically finite [F4], $T\text{-}\mathrm{m}\mathrm{o}\mathrm{d}$ is a finite $k$-linear abelian category in the sense of Finite k-linear abelian categories; the equivalence of step 4.1, required for every $k$-linear abelian $\mathcal{E}$, is exactly the universal property of The Deligne product of finite linear categories, so $T\text{-}\mathrm{m}\mathrm{o}\mathrm{d}$ with $⊠$ is a Deligne product of $R\text{-}\mathrm{m}\mathrm{o}\mathrm{d}$ and $S\text{-}\mathrm{m}\mathrm{o}\mathrm{d}$. For abstract finite $k$-linear abelian categories $\mathcal{A},\mathcal{B}$, [F5] provides equivalences $\mathcal{A}\simeq R\text{-}\mathrm{m}\mathrm{o}\mathrm{d}$, $\mathcal{B}\simeq S\text{-}\mathrm{m}\mathrm{o}\mathrm{d}$ with $R,S$ finite-dimensional, and transporting $⊠$, the finite direct sums, the algebra $T$ and the universal property along them yields a Deligne product of $\mathcal{A}$ and $\mathcal{B}$, well defined up to an equivalence respecting the universal bifunctors as follows directly from the universal property: for two products $P,Q$, extend their universal bifunctors to right exact functors $P\to Q$ and $Q\to P$. The restrictions of both composites are isomorphic to the corresponding universal bifunctors, so full faithfulness of restriction lifts these isomorphisms to composites isomorphic to the identities. No commutativity of $R$ or $S$ was used. The construction inherits the set-indexed choices of [F3], made under the stated AC assumption.