It is not really categorical, so this is maybe more of a comment than an answer, but the way I find easiest to see that transfer really gives a homomorphism (independent of choice of coset representatives, but it's not clear to me that this issue is much easier from this viewpoint) is from a viewpoint which may be due to T. Yoshida, who wrote some papers on ``character-theoretic transfer" in the 70s. Given that $[G:H]$ is finite, consider a group homomorphism $\phi: H \to A$ where $A$ is an Abelian group. Let $R$ be the group ring ${\rm GF}(2)A.$ Consider $\phi$ as a rank $1$-representation of $H$ over $R$. Induce that to a representation from $G \to {\rm GL}_d(R),$ where $d = [G:H]$, and take the determinant of that induced representation. In the case that $A = H/H^{\prime}$, we (implicitly) obtain the homomorphism $V_G: G^{ab} \to H^{ab}.$.

It is not really categorical, so this is maybe more of a comment than an answer, but the way I find easiest to see that transfer really gives a homomorphism (independent of choice of coset representatives, but it's not clear to me that this issue is much easier from this viewpoint) is from a viewpoint which may be due to T. Yoshida, who wrote some papers on "character-theoretic transfer" in the 70s. Given that $[G:H]$ is finite, consider a group homomorphism $\phi: H \to A$ where $A$ is an Abelian group. Let $R$ be the group ring ${\rm GF}(2)[A].$ Consider $\phi$ as a rank $1$-representation of $H$ over $R$. Induce that to a representation from $G \to {\rm GL}_d(R),$ where $d = [G:H]$, and take the determinant of that induced representation. In the case that $A = H/H^{\prime}$, we (implicitly) obtain the homomorphism $V_G: G^{ab} \to H^{ab}.$.