There obviously seems to be a connection between the push-forward and the pull-back of a smooth function $f:M \to N$ between smooth manifolds, and the Hom-maps from category theory $f^*=Mor_C(f,\mathbb{R})$ and $f_*=Mor_C(f,\mathbb{R})$ or some other Hom-maps, where $C$ is the category of differentiable manifolds with differentiable maps.
Since I'm not very good at category theory and the whole thing seems a bit intricate to me, I can't really figure it out. Any thoughts/ more precise statement on that topic would be very helpful