Concordance maps in $HFK^{-}$

We show that a decorated knot concordance $\mathcal{C}$ from $K_0$ to $K_1$ induces an $\mathbb{F}[U]$-module homomorphism $$G_{\mathcal{C}}: HFK^{-}(-S^3,K_0) \to HFK^{-}(-S^3,K_1)$$ which preserves the Alexander and absolute $\mathbb{Z}_2$-Maslov gradings. Our construction generalizes the concordance maps induced on $\widehat{HFK}$ studied by Juh\'asz and Marengon, but uses the description of $HFK^{-}$ as a direct limit of maps between sutured Floer homology groups discovered by Etnyre, Vela-Vick, and Zarev.