Let $f: X \rightarrow Y$ be continuous, X,Y pathwise connected and aspherical (i.e. trivial hight homotopy groups). Then $\pi_1(X)$ acts on the universal cover of $X$ via deck transformations, and on the universal cover of $Y$ via

$$([\sigma],y)\mapsto [\sigma \circ f]y$$

If g is another continuous map whose lift to the universal covers is equivariant under these actions, is g homotopic to f?

Let $f: X \rightarrow Y$ be continuous, X,Y pathwise connected and aspherical (i.e. trivial higher homotopy groups). Then $\pi_1(X)$ acts on the universal cover of $X$ via deck transformations, and on the universal cover of $Y$ via

$$([\sigma],y)\mapsto [\sigma \circ f]y$$

If g is another continuous map whose lift to the universal covers is equivariant under these actions, is g homotopic to f?