Lovász's homomorphism matrix lemma. Every homomorphism $F i →g F j$ factors as a strongly surjective homomorphism onto its image followed by an injective one; the image is isomorphic to a unique member $F k$ of the family, and each homomorphism admits exactly $aut(F k)$ factorisations through $F k$. Counting gives $M = S · D⁻¹ · I$ with $S$ the matrix of strongly surjective counts, $D$ the diagonal of automorphism counts and $I$ the matrix of injective counts. Ordering the family by number of vertices, then by number of edges, makes $S$ lower and $I$ upper triangular, both with positive diagonal, so all three factors are invertible and hence so is $M$.