A determined linear combination determines its constituents. Enlarge the family to a family $M$ of representatives of all graphs on at most $n$ vertices, $n$ being the bound on the sizes of the members, and extend the coefficients by zero. Multiplying the hypothesis by $hom(-, M k)$ for each $k$ — legitimate because $R$ is preserved under categorical products — turns it into the statement that a single vector meets the homomorphism matrix of $M$ in the same way for the two graphs. That matrix is invertible, so the vectors agree coordinatewise, and dividing by the nonzero coefficient gives the claim.