Let $M$ and $N$ be manifolds of same dimension with boundary. Let $f \colon M \to N$ be a continuous map. Apparently if the diagram $\require{AMScd}$ \begin{CD} \partial M @>>>M\\ @VVV @VVV{f}\\ \partial N @>>> N \end{CD}
commutes up to homotopy, then there is a map $\tilde f \colon M \to N$ homotopic to $f$ such that the diagram commutes strict. I do not see why this is true.