Straight homotopy invariants

Let $X$ and $Y$ be spaces and $M$ be an abelian group. A homotopy invariant $f\text{\hspace{0.17em}⁣}:[X,Y]\to M$ is called straight if there exists a homomorphism $F\colon L(X,Y)\to M$suchthat$f([a])=F(\langle a\rangle)$forall$a\in C(X,Y)$\mathrm{.}Here$\langle a\rangle\colon\langle X\rangle\to\langle Y\rangle$ is the homomorphism induced by $a$ between the abelian groups freely generated by $X$ and $Y$ and $L(X,Y)$ is a certain group of admissible' homomorphisms. We show that all straight invariants can be expressed through a universal’ straight invariant of homological nature.